Distance on a great circle

Percentage Accurate: 62.3% → 98.7%

Time: 36.2s

Alternatives: 45

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0))))
   (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	return R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0)
    code = r * (2.0d0 * atan2(sqrt(t_1), sqrt((1.0d0 - t_1))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_1 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * t_0) * t_0);
	return R * (2.0 * Math.atan2(Math.sqrt(t_1), Math.sqrt((1.0 - t_1))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0)
	return R * (2.0 * math.atan2(math.sqrt(t_1), math.sqrt((1.0 - t_1))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = (sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Initial Program: 62.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0))))
   (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	return R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0)
    code = r * (2.0d0 * atan2(sqrt(t_1), sqrt((1.0d0 - t_1))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_1 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * t_0) * t_0);
	return R * (2.0 * Math.atan2(Math.sqrt(t_1), Math.sqrt((1.0 - t_1))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0)
	return R * (2.0 * math.atan2(math.sqrt(t_1), math.sqrt((1.0 - t_1))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = (sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Alternative 1: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot 0.5\right), \cos \left(\pi \cdot 0.5\right), \cos \left(-0.5 \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (*
           (cos phi2)
           (pow
            (-
             (*
              (fma
               (sin (* lambda2 0.5))
               (cos (* PI 0.5))
               (* (cos (* -0.5 lambda2)) (sin (* PI 0.5))))
              (sin (* 0.5 lambda1)))
             (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
            2.0))
          (pow
           (fma
            (cos (* -0.5 phi2))
            (sin (* 0.5 phi1))
            (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
           2.0))))
   (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi1), (cos(phi2) * pow(((fma(sin((lambda2 * 0.5)), cos((((double) M_PI) * 0.5)), (cos((-0.5 * lambda2)) * sin((((double) M_PI) * 0.5)))) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0)), pow(fma(cos((-0.5 * phi2)), sin((0.5 * phi1)), (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0));
	return R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi1), Float64(cos(phi2) * (Float64(Float64(fma(sin(Float64(lambda2 * 0.5)), cos(Float64(pi * 0.5)), Float64(cos(Float64(-0.5 * lambda2)) * sin(Float64(pi * 0.5)))) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0)), (fma(cos(Float64(-0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[(N[(N[Sin[N[(lambda2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 63.2%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
4. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Applied rewrites 78.5%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
6. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. div-sub N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. sin-diff N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. lower-*.f64 77.5

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

7. Applied rewrites 77.5%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
8. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. div-sub N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. sin-diff N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. lower-*.f64 79.1

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

9. Applied rewrites 79.1%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
10. Step-by-step derivation

    1. lift-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    2. lift-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    3. lift--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    4. div-sub N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    5. sin-diff N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    6. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    7. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    8. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    9. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    10. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    11. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    12. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    13. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    14. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    15. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    16. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    17. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    18. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    19. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    20. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    21. lower-sin.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    22. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    23. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    24. lower-*.f64 78.7

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

11. Applied rewrites 78.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
12. Step-by-step derivation

    1. lift-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

    2. lift-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

    3. lift--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

    4. div-sub N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

    5. sin-diff N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

    6. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

    7. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    8. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    9. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    10. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    11. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    12. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    13. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    14. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    15. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    16. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

    17. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    18. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    19. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    20. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    21. lower-sin.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

    22. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

    23. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

    24. lower-*.f64 98.7

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

13. Applied rewrites 98.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

14. Taylor expanded in lambda1 around inf

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
15. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

16. Applied rewrites 98.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

17. Taylor expanded in lambda1 around inf

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
18. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

19. Applied rewrites 98.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]
20. Step-by-step derivation

    1. lift-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    2. sin-+PI/2-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\sin \left(\frac{1}{2} \cdot \lambda_2 + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    3. sin-sum N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\left(\sin \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    4. lift-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\left(\sin \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    5. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \lambda_2\right), \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    6. lift-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \lambda_2\right), \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    7. *-commutative N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    8. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    9. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    10. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    11. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    12. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    13. lower-PI.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    14. lift-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    15. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

21. Applied rewrites 98.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot 0.5\right), \cos \left(\pi \cdot 0.5\right), \cos \left(-0.5 \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
22. Step-by-step derivation

    1. lift-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    2. sin-+PI/2-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\sin \left(\frac{1}{2} \cdot \lambda_2 + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    3. sin-sum N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\left(\sin \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    4. lift-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\left(\sin \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    5. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \lambda_2\right), \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    6. lift-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \lambda_2\right), \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    7. *-commutative N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    8. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    9. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    10. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    11. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    12. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\mathsf{PI}\left(\right) \cdot \frac{1}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    13. lower-PI.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    14. lift-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

    15. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\pi \cdot \frac{1}{2}\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

23. Applied rewrites 98.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot 0.5\right), \cos \left(\pi \cdot 0.5\right), \cos \left(-0.5 \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\sin \left(\lambda_2 \cdot 0.5\right), \cos \left(\pi \cdot 0.5\right), \cos \left(-0.5 \cdot \lambda_2\right) \cdot \sin \left(\pi \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
24. Add Preprocessing

Alternative 2: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (*
           (cos phi2)
           (pow
            (-
             (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
             (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
            2.0))
          (pow
           (fma
            (cos (* -0.5 phi2))
            (sin (* 0.5 phi1))
            (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
           2.0))))
   (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi1), (cos(phi2) * pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0)), pow(fma(cos((-0.5 * phi2)), sin((0.5 * phi1)), (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0));
	return R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi1), Float64(cos(phi2) * (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0)), (fma(cos(Float64(-0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 63.2%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
4. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Applied rewrites 78.5%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
6. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. div-sub N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. sin-diff N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. lower-*.f64 77.5

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

7. Applied rewrites 77.5%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
8. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. div-sub N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. sin-diff N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. lower-*.f64 79.1

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

9. Applied rewrites 79.1%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
10. Step-by-step derivation

    1. lift-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    2. lift-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    3. lift--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    4. div-sub N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    5. sin-diff N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    6. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    7. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    8. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    9. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    10. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    11. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    12. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    13. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    14. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    15. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    16. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    17. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    18. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    19. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    20. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    21. lower-sin.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    22. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    23. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

    24. lower-*.f64 78.7

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

11. Applied rewrites 78.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
12. Step-by-step derivation

    1. lift-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

    2. lift-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

    3. lift--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

    4. div-sub N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

    5. sin-diff N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

    6. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

    7. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    8. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    9. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    10. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    11. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    12. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    13. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    14. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    15. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    16. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

    17. lower-cos.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    18. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    19. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    20. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

    21. lower-sin.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

    22. mult-flip N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

    23. metadata-eval N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

    24. lower-*.f64 98.7

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

13. Applied rewrites 98.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

14. Taylor expanded in lambda1 around inf

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
15. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

16. Applied rewrites 98.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

17. Taylor expanded in lambda1 around inf

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
18. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

19. Applied rewrites 98.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]
20. Add Preprocessing

Alternative 3: 88.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + -0.125 \cdot {\phi_2}^{2}, t\_0, t\_1\right)\right)}^{2}\right)\\ t_3 := \cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\\ t_4 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(1 + \frac{\cos \phi_2 \cdot 0.5}{t\_3}\right) \cdot t\_3\right)\\ t_5 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), t\_0, t\_1\right)\right)}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -550000000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1 (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
        (t_2
         (fma
          (cos phi1)
          (*
           (cos phi2)
           (pow
            (-
             (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
             (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
            2.0))
          (pow (fma (+ 1.0 (* -0.125 (pow phi2 2.0))) t_0 t_1) 2.0)))
        (t_3 (* (cos phi2) (* (cos (- lambda2 lambda1)) -0.5)))
        (t_4
         (+
          (pow
           (fma
            (sin (* phi1 0.5))
            (cos (/ phi2 -2.0))
            (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
           2.0)
          (* (cos phi1) (* (+ 1.0 (/ (* (cos phi2) 0.5) t_3)) t_3))))
        (t_5
         (fma
          (cos phi1)
          (* (cos phi2) (/ 1.0 (pow (sin (* (- lambda2 lambda1) -0.5)) -2.0)))
          (pow (fma (cos (* -0.5 phi2)) t_0 t_1) 2.0))))
   (if (<= phi2 -550000000000.0)
     (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1));
	double t_1 = cos((0.5 * phi1)) * sin((-0.5 * phi2));
	double t_2 = fma(cos(phi1), (cos(phi2) * pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0)), pow(fma((1.0 + (-0.125 * pow(phi2, 2.0))), t_0, t_1), 2.0));
	double t_3 = cos(phi2) * (cos((lambda2 - lambda1)) * -0.5);
	double t_4 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + (cos(phi1) * ((1.0 + ((cos(phi2) * 0.5) / t_3)) * t_3));
	double t_5 = fma(cos(phi1), (cos(phi2) * (1.0 / pow(sin(((lambda2 - lambda1) * -0.5)), -2.0))), pow(fma(cos((-0.5 * phi2)), t_0, t_1), 2.0));
	double tmp;
	if (phi2 <= -550000000000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi1))
	t_1 = Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))
	t_2 = fma(cos(phi1), Float64(cos(phi2) * (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0)), (fma(Float64(1.0 + Float64(-0.125 * (phi2 ^ 2.0))), t_0, t_1) ^ 2.0))
	t_3 = Float64(cos(phi2) * Float64(cos(Float64(lambda2 - lambda1)) * -0.5))
	t_4 = Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + Float64(cos(phi1) * Float64(Float64(1.0 + Float64(Float64(cos(phi2) * 0.5) / t_3)) * t_3)))
	t_5 = fma(cos(phi1), Float64(cos(phi2) * Float64(1.0 / (sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ -2.0))), (fma(cos(Float64(-0.5 * phi2)), t_0, t_1) ^ 2.0))
	tmp = 0.0
	if (phi2 <= -550000000000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(1.0 + N[(-0.125 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(1.0 + N[(N[(N[Cos[phi2], $MachinePrecision] * 0.5), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(1.0 / N[Power[N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -550000000000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -5.5e11

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 79.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{\color{blue}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)}^{-2}}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 78.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{\color{blue}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   if -5.5e11 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   20. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   21. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       3. lower-pow.f64 63.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + -0.125 \cdot {\phi_2}^{2}, \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   22. Applied rewrites 63.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + -0.125 \cdot {\phi_2}^{2}, \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   23. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   24. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       3. lower-pow.f64 54.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + -0.125 \cdot {\phi_2}^{2}, \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + -0.125 \cdot {\phi_2}^{2}, \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   25. Applied rewrites 54.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + -0.125 \cdot {\phi_2}^{2}, \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + -0.125 \cdot {\phi_2}^{2}, \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   if 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) + \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. distribute-lft-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) + \cos \phi_2 \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. sum-to-mult N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(1 + \frac{\cos \phi_2 \cdot \frac{1}{2}}{\cos \phi_2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right) \cdot \left(\cos \phi_2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower-unsound-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(1 + \frac{\cos \phi_2 \cdot \frac{1}{2}}{\cos \phi_2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right) \cdot \left(\cos \phi_2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 75.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(1 + \frac{\cos \phi_2 \cdot 0.5}{\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(1 + \frac{\cos \phi_2 \cdot \frac{1}{2}}{\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)}\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}\right)}}\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(1 + \frac{\cos \phi_2 \cdot \frac{1}{2}}{\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)}\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) + \frac{1}{2}\right)}\right)\right)}}\right) \]

       3. distribute-lft-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(1 + \frac{\cos \phi_2 \cdot \frac{1}{2}}{\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)}\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) + \cos \phi_2 \cdot \frac{1}{2}\right)}\right)}}\right) \]

       4. sum-to-mult N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(1 + \frac{\cos \phi_2 \cdot \frac{1}{2}}{\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)}\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(1 + \frac{\cos \phi_2 \cdot \frac{1}{2}}{\cos \phi_2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right) \cdot \left(\cos \phi_2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\right)}}\right) \]

       5. lower-unsound-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(1 + \frac{\cos \phi_2 \cdot \frac{1}{2}}{\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)}\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(1 + \frac{\cos \phi_2 \cdot \frac{1}{2}}{\cos \phi_2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right) \cdot \left(\cos \phi_2 \cdot \left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\right)}}\right) \]

   13. Applied rewrites 75.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(1 + \frac{\cos \phi_2 \cdot 0.5}{\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(1 + \frac{\cos \phi_2 \cdot 0.5}{\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}\right) \cdot \left(\cos \phi_2 \cdot \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right)}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + -0.125 \cdot {\phi_2}^{2}, t\_0, t\_1\right)\right)}^{2}\right)\\ t_3 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)\\ t_4 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), t\_0, t\_1\right)\right)}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -550000000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1 (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
        (t_2
         (fma
          (cos phi1)
          (*
           (cos phi2)
           (pow
            (-
             (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
             (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
            2.0))
          (pow (fma (+ 1.0 (* -0.125 (pow phi2 2.0))) t_0 t_1) 2.0)))
        (t_3
         (+
          (pow
           (fma
            (sin (* phi1 0.5))
            (cos (/ phi2 -2.0))
            (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
           2.0)
          (*
           (cos phi1)
           (-
            (* (* (cos (- lambda2 lambda1)) -0.5) (cos phi2))
            (* -0.5 (cos phi2))))))
        (t_4
         (fma
          (cos phi1)
          (* (cos phi2) (/ 1.0 (pow (sin (* (- lambda2 lambda1) -0.5)) -2.0)))
          (pow (fma (cos (* -0.5 phi2)) t_0 t_1) 2.0))))
   (if (<= phi2 -550000000000.0)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1));
	double t_1 = cos((0.5 * phi1)) * sin((-0.5 * phi2));
	double t_2 = fma(cos(phi1), (cos(phi2) * pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0)), pow(fma((1.0 + (-0.125 * pow(phi2, 2.0))), t_0, t_1), 2.0));
	double t_3 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + (cos(phi1) * (((cos((lambda2 - lambda1)) * -0.5) * cos(phi2)) - (-0.5 * cos(phi2))));
	double t_4 = fma(cos(phi1), (cos(phi2) * (1.0 / pow(sin(((lambda2 - lambda1) * -0.5)), -2.0))), pow(fma(cos((-0.5 * phi2)), t_0, t_1), 2.0));
	double tmp;
	if (phi2 <= -550000000000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi1))
	t_1 = Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))
	t_2 = fma(cos(phi1), Float64(cos(phi2) * (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0)), (fma(Float64(1.0 + Float64(-0.125 * (phi2 ^ 2.0))), t_0, t_1) ^ 2.0))
	t_3 = Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + Float64(cos(phi1) * Float64(Float64(Float64(cos(Float64(lambda2 - lambda1)) * -0.5) * cos(phi2)) - Float64(-0.5 * cos(phi2)))))
	t_4 = fma(cos(phi1), Float64(cos(phi2) * Float64(1.0 / (sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ -2.0))), (fma(cos(Float64(-0.5 * phi2)), t_0, t_1) ^ 2.0))
	tmp = 0.0
	if (phi2 <= -550000000000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(1.0 + N[(-0.125 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(1.0 / N[Power[N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -550000000000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -5.5e11

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 79.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{\color{blue}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)}^{-2}}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 78.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{\color{blue}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   if -5.5e11 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   20. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   21. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       3. lower-pow.f64 63.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + -0.125 \cdot {\phi_2}^{2}, \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   22. Applied rewrites 63.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + -0.125 \cdot {\phi_2}^{2}, \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   23. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   24. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + \frac{-1}{8} \cdot {\phi_2}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       3. lower-pow.f64 54.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + -0.125 \cdot {\phi_2}^{2}, \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + -0.125 \cdot {\phi_2}^{2}, \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   25. Applied rewrites 54.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + -0.125 \cdot {\phi_2}^{2}, \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(1 + -0.125 \cdot {\phi_2}^{2}, \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   if 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) + \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 + \frac{1}{2} \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. fp-cancel-sign-sub-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       9. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       10. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       11. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       15. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       16. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       17. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       18. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       19. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       20. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       21. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \color{blue}{\frac{-1}{2}} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - \color{blue}{-0.5 \cdot \cos \phi_2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

   11. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}\right)}}\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) + \frac{1}{2}\right)}\right)\right)}}\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 + \frac{1}{2} \cdot \cos \phi_2\right)}\right)}}\right) \]

       4. fp-cancel-sign-sub-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       8. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       9. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       10. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       11. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       12. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       15. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       16. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       17. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       18. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       19. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       20. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       21. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \color{blue}{\frac{-1}{2}} \cdot \cos \phi_2\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - \color{blue}{-0.5 \cdot \cos \phi_2}\right)\right)}}\right) \]

   13. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 88.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2}\\ t_1 := \sin \left(-0.5 \cdot \phi_2\right)\\ t_2 := \cos \left(-0.5 \cdot \phi_2\right)\\ t_3 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_4 := t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, t\_3, 0.5\right)\right)\\ t_5 := {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}\\ t_6 := t\_0 + \cos \phi_1 \cdot \left(\left(t\_3 \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)\\ \mathbf{if}\;\phi_1 \leq -4.4 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{elif}\;\phi_1 \leq 0.92:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t\_5, {\left(\mathsf{fma}\left(t\_2, \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot t\_1\right)\right)}^{2}\right)}}{\sqrt{\left(1 + -1 \cdot \left(\phi_1 \cdot \left(t\_2 \cdot t\_1\right)\right)\right) - \mathsf{fma}\left(\cos \phi_2, t\_5, {t\_1}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_6}}{\sqrt{1 - t\_6}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (sin (* phi1 0.5))
           (cos (/ phi2 -2.0))
           (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
          2.0))
        (t_1 (sin (* -0.5 phi2)))
        (t_2 (cos (* -0.5 phi2)))
        (t_3 (cos (- lambda2 lambda1)))
        (t_4 (+ t_0 (* (cos phi1) (* (cos phi2) (fma -0.5 t_3 0.5)))))
        (t_5
         (pow
          (-
           (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
           (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
          2.0))
        (t_6
         (+
          t_0
          (* (cos phi1) (- (* (* t_3 -0.5) (cos phi2)) (* -0.5 (cos phi2)))))))
   (if (<= phi1 -4.4e-8)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (if (<= phi1 0.92)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (* (cos phi2) t_5)
            (pow (fma t_2 (sin (* 0.5 phi1)) (* (cos (* 0.5 phi1)) t_1)) 2.0)))
          (sqrt
           (-
            (+ 1.0 (* -1.0 (* phi1 (* t_2 t_1))))
            (fma (cos phi2) t_5 (pow t_1 2.0)))))))
       (* R (* 2.0 (atan2 (sqrt t_6) (sqrt (- 1.0 t_6)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0);
	double t_1 = sin((-0.5 * phi2));
	double t_2 = cos((-0.5 * phi2));
	double t_3 = cos((lambda2 - lambda1));
	double t_4 = t_0 + (cos(phi1) * (cos(phi2) * fma(-0.5, t_3, 0.5)));
	double t_5 = pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0);
	double t_6 = t_0 + (cos(phi1) * (((t_3 * -0.5) * cos(phi2)) - (-0.5 * cos(phi2))));
	double tmp;
	if (phi1 <= -4.4e-8) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else if (phi1 <= 0.92) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * t_5), pow(fma(t_2, sin((0.5 * phi1)), (cos((0.5 * phi1)) * t_1)), 2.0))), sqrt(((1.0 + (-1.0 * (phi1 * (t_2 * t_1)))) - fma(cos(phi2), t_5, pow(t_1, 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_6), sqrt((1.0 - t_6))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0
	t_1 = sin(Float64(-0.5 * phi2))
	t_2 = cos(Float64(-0.5 * phi2))
	t_3 = cos(Float64(lambda2 - lambda1))
	t_4 = Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * fma(-0.5, t_3, 0.5))))
	t_5 = Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0
	t_6 = Float64(t_0 + Float64(cos(phi1) * Float64(Float64(Float64(t_3 * -0.5) * cos(phi2)) - Float64(-0.5 * cos(phi2)))))
	tmp = 0.0
	if (phi1 <= -4.4e-8)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	elseif (phi1 <= 0.92)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * t_5), (fma(t_2, sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * t_1)) ^ 2.0))), sqrt(Float64(Float64(1.0 + Float64(-1.0 * Float64(phi1 * Float64(t_2 * t_1)))) - fma(cos(phi2), t_5, (t_1 ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_6), sqrt(Float64(1.0 - t_6)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * t$95$3 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[(t$95$3 * -0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -4.4e-8], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.92], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$5), $MachinePrecision] + N[Power[N[(t$95$2 * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 + N[(-1.0 * N[(phi1 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * t$95$5 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$6], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$6), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -4.3999999999999997e-8

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   if -4.3999999999999997e-8 < phi1 < 0.92000000000000004

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   20. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{\color{blue}{\left(1 + -1 \cdot \left(\phi_1 \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)\right) - \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   21. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{\left(1 + -1 \cdot \left(\phi_1 \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)\right) - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   22. Applied rewrites 53.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{\color{blue}{\left(1 + -1 \cdot \left(\phi_1 \cdot \left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)\right) - \mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if 0.92000000000000004 < phi1

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) + \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 + \frac{1}{2} \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. fp-cancel-sign-sub-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       9. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       10. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       11. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       15. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       16. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       17. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       18. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       19. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       20. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       21. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \color{blue}{\frac{-1}{2}} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - \color{blue}{-0.5 \cdot \cos \phi_2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

   11. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}\right)}}\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) + \frac{1}{2}\right)}\right)\right)}}\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 + \frac{1}{2} \cdot \cos \phi_2\right)}\right)}}\right) \]

       4. fp-cancel-sign-sub-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       8. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       9. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       10. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       11. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       12. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       15. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       16. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       17. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       18. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       19. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       20. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       21. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \color{blue}{\frac{-1}{2}} \cdot \cos \phi_2\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - \color{blue}{-0.5 \cdot \cos \phi_2}\right)\right)}}\right) \]

   13. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 88.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2}\\ t_1 := \sin \left(-0.5 \cdot \phi_2\right)\\ t_2 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_3 := t\_0 + \cos \phi_1 \cdot \left(\left(t\_2 \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)\\ t_4 := t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, t\_2, 0.5\right)\right)\\ t_5 := {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -3.4 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{elif}\;\phi_1 \leq 216:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t\_5, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot t\_1\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, t\_5, {t\_1}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (sin (* phi1 0.5))
           (cos (/ phi2 -2.0))
           (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
          2.0))
        (t_1 (sin (* -0.5 phi2)))
        (t_2 (cos (- lambda2 lambda1)))
        (t_3
         (+
          t_0
          (* (cos phi1) (- (* (* t_2 -0.5) (cos phi2)) (* -0.5 (cos phi2))))))
        (t_4 (+ t_0 (* (cos phi1) (* (cos phi2) (fma -0.5 t_2 0.5)))))
        (t_5
         (pow
          (-
           (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
           (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
          2.0)))
   (if (<= phi1 -3.4e-15)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (if (<= phi1 216.0)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (* (cos phi2) t_5)
            (pow
             (fma
              (cos (* -0.5 phi2))
              (sin (* 0.5 phi1))
              (* (cos (* 0.5 phi1)) t_1))
             2.0)))
          (sqrt (- 1.0 (fma (cos phi2) t_5 (pow t_1 2.0)))))))
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0);
	double t_1 = sin((-0.5 * phi2));
	double t_2 = cos((lambda2 - lambda1));
	double t_3 = t_0 + (cos(phi1) * (((t_2 * -0.5) * cos(phi2)) - (-0.5 * cos(phi2))));
	double t_4 = t_0 + (cos(phi1) * (cos(phi2) * fma(-0.5, t_2, 0.5)));
	double t_5 = pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0);
	double tmp;
	if (phi1 <= -3.4e-15) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else if (phi1 <= 216.0) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * t_5), pow(fma(cos((-0.5 * phi2)), sin((0.5 * phi1)), (cos((0.5 * phi1)) * t_1)), 2.0))), sqrt((1.0 - fma(cos(phi2), t_5, pow(t_1, 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0
	t_1 = sin(Float64(-0.5 * phi2))
	t_2 = cos(Float64(lambda2 - lambda1))
	t_3 = Float64(t_0 + Float64(cos(phi1) * Float64(Float64(Float64(t_2 * -0.5) * cos(phi2)) - Float64(-0.5 * cos(phi2)))))
	t_4 = Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * fma(-0.5, t_2, 0.5))))
	t_5 = Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0
	tmp = 0.0
	if (phi1 <= -3.4e-15)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	elseif (phi1 <= 216.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * t_5), (fma(cos(Float64(-0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * t_1)) ^ 2.0))), sqrt(Float64(1.0 - fma(cos(phi2), t_5, (t_1 ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[(t$95$2 * -0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * t$95$2 + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[phi1, -3.4e-15], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 216.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$5), $MachinePrecision] + N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi2], $MachinePrecision] * t$95$5 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.4e-15

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   if -3.4e-15 < phi1 < 216

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   20. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   21. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   22. Applied rewrites 59.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if 216 < phi1

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) + \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 + \frac{1}{2} \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. fp-cancel-sign-sub-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       9. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       10. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       11. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       15. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       16. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       17. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       18. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       19. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       20. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       21. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \color{blue}{\frac{-1}{2}} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - \color{blue}{-0.5 \cdot \cos \phi_2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

   11. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}\right)}}\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) + \frac{1}{2}\right)}\right)\right)}}\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 + \frac{1}{2} \cdot \cos \phi_2\right)}\right)}}\right) \]

       4. fp-cancel-sign-sub-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       8. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       9. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       10. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       11. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       12. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       15. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       16. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       17. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       18. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       19. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       20. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       21. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \color{blue}{\frac{-1}{2}} \cdot \cos \phi_2\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - \color{blue}{-0.5 \cdot \cos \phi_2}\right)\right)}}\right) \]

   13. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 88.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}\\ t_2 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)\\ t_3 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), t\_0, \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\\ t_4 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}, t\_3\right)\\ \mathbf{if}\;\phi_2 \leq -7 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t\_1, t\_3\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_1, {t\_0}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1
         (pow
          (-
           (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
           (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
          2.0))
        (t_2
         (+
          (pow
           (fma
            (sin (* phi1 0.5))
            (cos (/ phi2 -2.0))
            (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
           2.0)
          (*
           (cos phi1)
           (-
            (* (* (cos (- lambda2 lambda1)) -0.5) (cos phi2))
            (* -0.5 (cos phi2))))))
        (t_3
         (pow
          (fma
           (cos (* -0.5 phi2))
           t_0
           (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
          2.0))
        (t_4
         (fma
          (cos phi1)
          (* (cos phi2) (/ 1.0 (pow (sin (* (- lambda2 lambda1) -0.5)) -2.0)))
          t_3)))
   (if (<= phi2 -7e-11)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (if (<= phi2 7.5e-20)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (fma (cos phi1) (* (cos phi2) t_1) t_3))
          (sqrt (- 1.0 (fma (cos phi1) t_1 (pow t_0 2.0)))))))
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1));
	double t_1 = pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0);
	double t_2 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + (cos(phi1) * (((cos((lambda2 - lambda1)) * -0.5) * cos(phi2)) - (-0.5 * cos(phi2))));
	double t_3 = pow(fma(cos((-0.5 * phi2)), t_0, (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0);
	double t_4 = fma(cos(phi1), (cos(phi2) * (1.0 / pow(sin(((lambda2 - lambda1) * -0.5)), -2.0))), t_3);
	double tmp;
	if (phi2 <= -7e-11) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * t_1), t_3)), sqrt((1.0 - fma(cos(phi1), t_1, pow(t_0, 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi1))
	t_1 = Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0
	t_2 = Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + Float64(cos(phi1) * Float64(Float64(Float64(cos(Float64(lambda2 - lambda1)) * -0.5) * cos(phi2)) - Float64(-0.5 * cos(phi2)))))
	t_3 = fma(cos(Float64(-0.5 * phi2)), t_0, Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0
	t_4 = fma(cos(phi1), Float64(cos(phi2) * Float64(1.0 / (sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ -2.0))), t_3)
	tmp = 0.0
	if (phi2 <= -7e-11)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * t_1), t_3)), sqrt(Float64(1.0 - fma(cos(phi1), t_1, (t_0 ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$0 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(1.0 / N[Power[N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[phi2, -7e-11], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * t$95$1 + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -7.00000000000000038e-11

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 79.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{\color{blue}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)}^{-2}}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 78.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{\color{blue}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   if -7.00000000000000038e-11 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   20. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   21. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   22. Applied rewrites 59.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) + \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 + \frac{1}{2} \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. fp-cancel-sign-sub-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       9. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       10. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       11. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       15. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       16. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       17. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       18. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       19. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       20. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       21. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \color{blue}{\frac{-1}{2}} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - \color{blue}{-0.5 \cdot \cos \phi_2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

   11. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}\right)}}\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) + \frac{1}{2}\right)}\right)\right)}}\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 + \frac{1}{2} \cdot \cos \phi_2\right)}\right)}}\right) \]

       4. fp-cancel-sign-sub-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       8. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       9. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       10. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       11. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       12. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       15. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       16. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       17. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       18. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       19. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       20. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       21. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \color{blue}{\frac{-1}{2}} \cdot \cos \phi_2\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - \color{blue}{-0.5 \cdot \cos \phi_2}\right)\right)}}\right) \]

   13. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {t\_0}^{2}\right)\\ t_2 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), t\_0, \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -1.55 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow t_0 2.0)))
        (t_2
         (+
          (pow
           (fma
            (sin (* phi1 0.5))
            (cos (/ phi2 -2.0))
            (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
           2.0)
          (*
           (cos phi1)
           (-
            (* (* (cos (- lambda2 lambda1)) -0.5) (cos phi2))
            (* -0.5 (cos phi2))))))
        (t_3
         (fma
          (cos phi1)
          (* (cos phi2) (/ 1.0 (pow (sin (* (- lambda2 lambda1) -0.5)) -2.0)))
          (pow
           (fma
            (cos (* -0.5 phi2))
            t_0
            (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
           2.0))))
   (if (<= phi2 -1.55e-18)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1));
	double t_1 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(t_0, 2.0));
	double t_2 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + (cos(phi1) * (((cos((lambda2 - lambda1)) * -0.5) * cos(phi2)) - (-0.5 * cos(phi2))));
	double t_3 = fma(cos(phi1), (cos(phi2) * (1.0 / pow(sin(((lambda2 - lambda1) * -0.5)), -2.0))), pow(fma(cos((-0.5 * phi2)), t_0, (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0));
	double tmp;
	if (phi2 <= -1.55e-18) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi1))
	t_1 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (t_0 ^ 2.0))
	t_2 = Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + Float64(cos(phi1) * Float64(Float64(Float64(cos(Float64(lambda2 - lambda1)) * -0.5) * cos(phi2)) - Float64(-0.5 * cos(phi2)))))
	t_3 = fma(cos(phi1), Float64(cos(phi2) * Float64(1.0 / (sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ -2.0))), (fma(cos(Float64(-0.5 * phi2)), t_0, Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0))
	tmp = 0.0
	if (phi2 <= -1.55e-18)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] - N[(-0.5 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(1.0 / N[Power[N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$0 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.55e-18], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.55000000000000003e-18

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 79.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{\color{blue}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)}^{-2}}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 78.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{\color{blue}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   if -1.55000000000000003e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) + \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 + \frac{1}{2} \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. fp-cancel-sign-sub-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       9. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       10. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       11. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       15. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       16. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       17. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       18. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       19. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       20. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       21. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \color{blue}{\frac{-1}{2}} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - \color{blue}{-0.5 \cdot \cos \phi_2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

   11. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}\right)}}\right) \]

       2. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right) + \frac{1}{2}\right)}\right)\right)}}\right) \]

       3. distribute-rgt-in N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 + \frac{1}{2} \cdot \cos \phi_2\right)}\right)}}\right) \]

       4. fp-cancel-sign-sub-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \cos \phi_2} - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       8. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       9. cos-neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       10. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       11. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       12. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \frac{-1}{2}\right)} \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       15. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       16. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       17. sub-negate-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       18. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 - \lambda_1\right)}\right)\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       19. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       20. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \phi_2\right)\right)}}\right) \]

       21. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \frac{-1}{2} \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right) \cdot \cos \phi_2 - \color{blue}{\frac{-1}{2}} \cdot \cos \phi_2\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - \color{blue}{-0.5 \cdot \cos \phi_2}\right)\right)}}\right) \]

   13. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot -0.5\right) \cdot \cos \phi_2 - -0.5 \cdot \cos \phi_2\right)}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {t\_0}^{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), t\_0, \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -1.55 \cdot 10^{-18}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow t_0 2.0)))
        (t_2
         (fma
          (cos phi1)
          (* (cos phi2) (/ 1.0 (pow (sin (* (- lambda2 lambda1) -0.5)) -2.0)))
          (pow
           (fma
            (cos (* -0.5 phi2))
            t_0
            (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
           2.0)))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -1.55e-18)
     t_3
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1));
	double t_1 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(t_0, 2.0));
	double t_2 = fma(cos(phi1), (cos(phi2) * (1.0 / pow(sin(((lambda2 - lambda1) * -0.5)), -2.0))), pow(fma(cos((-0.5 * phi2)), t_0, (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -1.55e-18) {
		tmp = t_3;
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi1))
	t_1 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (t_0 ^ 2.0))
	t_2 = fma(cos(phi1), Float64(cos(phi2) * Float64(1.0 / (sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ -2.0))), (fma(cos(Float64(-0.5 * phi2)), t_0, Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -1.55e-18)
		tmp = t_3;
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(1.0 / N[Power[N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$0 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.55e-18], t$95$3, If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.55000000000000003e-18 or 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]
   20. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   21. Applied rewrites 79.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{\color{blue}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
   22. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)}^{-2}}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   23. Applied rewrites 78.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \frac{1}{\color{blue}{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{-2}}}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   if -1.55000000000000003e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\\ t_2 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + t\_1\\ t_3 := {\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), t\_0 \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2} + t\_1\\ t_4 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {t\_0}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1
         (*
          (cos phi1)
          (* (cos phi2) (fma -0.5 (cos (- lambda2 lambda1)) 0.5))))
        (t_2
         (+
          (pow
           (fma
            (sin (* phi1 0.5))
            (cos (/ phi2 -2.0))
            (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
           2.0)
          t_1))
        (t_3
         (+
          (pow
           (fma
            (sin (* phi2 -0.5))
            (cos (* 0.5 phi1))
            (* t_0 (cos (* phi2 -0.5))))
           2.0)
          t_1))
        (t_4
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow t_0 2.0))))
   (if (<= phi2 -1.6e-18)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1));
	double t_1 = cos(phi1) * (cos(phi2) * fma(-0.5, cos((lambda2 - lambda1)), 0.5));
	double t_2 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + t_1;
	double t_3 = pow(fma(sin((phi2 * -0.5)), cos((0.5 * phi1)), (t_0 * cos((phi2 * -0.5)))), 2.0) + t_1;
	double t_4 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(t_0, 2.0));
	double tmp;
	if (phi2 <= -1.6e-18) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi1))
	t_1 = Float64(cos(phi1) * Float64(cos(phi2) * fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)))
	t_2 = Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + t_1)
	t_3 = Float64((fma(sin(Float64(phi2 * -0.5)), cos(Float64(0.5 * phi1)), Float64(t_0 * cos(Float64(phi2 * -0.5)))) ^ 2.0) + t_1)
	t_4 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (t_0 ^ 2.0))
	tmp = 0.0
	if (phi2 <= -1.6e-18)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(t$95$0 * N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.6e-18], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.6e-18

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\phi_2}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \color{blue}{\frac{-1}{2}}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{-1}{2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       17. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       19. lift-sin.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \color{blue}{\sin \left(\phi_1 \cdot 0.5\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

       20. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \color{blue}{\left(0.5 \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

       23. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\phi_2}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \color{blue}{\frac{-1}{2}}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{-1}{2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       17. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       19. lift-sin.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \color{blue}{\sin \left(\phi_1 \cdot 0.5\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

       20. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \color{blue}{\left(0.5 \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

       23. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

   if -1.6e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right)\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\\ t_3 := {\left(t\_1 \cdot \cos \left(\phi_2 \cdot 0.5\right) - t\_0 \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + t\_2\\ t_4 := {\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), t\_0, t\_1 \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2} + t\_2\\ t_5 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {t\_1}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 phi1)))
        (t_1 (sin (* 0.5 phi1)))
        (t_2
         (*
          (cos phi1)
          (* (cos phi2) (fma -0.5 (cos (- lambda2 lambda1)) 0.5))))
        (t_3
         (+
          (pow (- (* t_1 (cos (* phi2 0.5))) (* t_0 (sin (* phi2 0.5)))) 2.0)
          t_2))
        (t_4
         (+
          (pow (fma (sin (* phi2 -0.5)) t_0 (* t_1 (cos (* phi2 -0.5)))) 2.0)
          t_2))
        (t_5
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow t_1 2.0))))
   (if (<= phi2 -1.6e-18)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * phi1));
	double t_1 = sin((0.5 * phi1));
	double t_2 = cos(phi1) * (cos(phi2) * fma(-0.5, cos((lambda2 - lambda1)), 0.5));
	double t_3 = pow(((t_1 * cos((phi2 * 0.5))) - (t_0 * sin((phi2 * 0.5)))), 2.0) + t_2;
	double t_4 = pow(fma(sin((phi2 * -0.5)), t_0, (t_1 * cos((phi2 * -0.5)))), 2.0) + t_2;
	double t_5 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(t_1, 2.0));
	double tmp;
	if (phi2 <= -1.6e-18) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * phi1))
	t_1 = sin(Float64(0.5 * phi1))
	t_2 = Float64(cos(phi1) * Float64(cos(phi2) * fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)))
	t_3 = Float64((Float64(Float64(t_1 * cos(Float64(phi2 * 0.5))) - Float64(t_0 * sin(Float64(phi2 * 0.5)))) ^ 2.0) + t_2)
	t_4 = Float64((fma(sin(Float64(phi2 * -0.5)), t_0, Float64(t_1 * cos(Float64(phi2 * -0.5)))) ^ 2.0) + t_2)
	t_5 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (t_1 ^ 2.0))
	tmp = 0.0
	if (phi2 <= -1.6e-18)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[(t$95$1 * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$0 + N[(t$95$1 * N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.6e-18], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.6e-18

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\phi_2}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \color{blue}{\frac{-1}{2}}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{-1}{2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       17. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       19. lift-sin.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \color{blue}{\sin \left(\phi_1 \cdot 0.5\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

       20. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \color{blue}{\left(0.5 \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

       23. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\phi_2}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \color{blue}{\frac{-1}{2}}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{-1}{2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       17. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       19. lift-sin.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \color{blue}{\sin \left(\phi_1 \cdot 0.5\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

       20. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \color{blue}{\left(0.5 \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

       23. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

   if -1.6e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{-2}\right)} + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{-2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. sin-sum-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\phi_1 \cdot \frac{1}{2} + \frac{\phi_2}{-2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\color{blue}{\phi_1 \cdot \frac{1}{2}} + \frac{\phi_2}{-2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}} + \frac{\phi_2}{-2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       10. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\color{blue}{\frac{\phi_1}{2}} + \frac{\phi_2}{-2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       11. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1}{2} + \color{blue}{\frac{\phi_2}{-2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       12. frac-2neg N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1}{2} + \color{blue}{\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(-2\right)}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       13. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       14. div-add N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       15. sub-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       16. div-sub N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       17. sin-diff N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       18. lower--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\phi_2}{-2}\right)} + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\phi_2}{-2}\right)}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. sin-sum-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\phi_1 \cdot \frac{1}{2} + \frac{\phi_2}{-2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\color{blue}{\phi_1 \cdot \frac{1}{2}} + \frac{\phi_2}{-2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}} + \frac{\phi_2}{-2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       10. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\color{blue}{\frac{\phi_1}{2}} + \frac{\phi_2}{-2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       11. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1}{2} + \color{blue}{\frac{\phi_2}{-2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       12. frac-2neg N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1}{2} + \color{blue}{\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(-2\right)}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       13. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{2}}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       14. div-add N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       15. sub-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       16. div-sub N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       17. sin-diff N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       18. lower--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 75.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := {\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), t\_0 \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {t\_0}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1
         (+
          (pow
           (fma
            (sin (* phi2 -0.5))
            (cos (* 0.5 phi1))
            (* t_0 (cos (* phi2 -0.5))))
           2.0)
          (*
           (cos phi1)
           (* (cos phi2) (fma -0.5 (cos (- lambda2 lambda1)) 0.5)))))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1))))))
        (t_3
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow t_0 2.0))))
   (if (<= phi2 -1.6e-18)
     t_2
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1));
	double t_1 = pow(fma(sin((phi2 * -0.5)), cos((0.5 * phi1)), (t_0 * cos((phi2 * -0.5)))), 2.0) + (cos(phi1) * (cos(phi2) * fma(-0.5, cos((lambda2 - lambda1)), 0.5)));
	double t_2 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double t_3 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(t_0, 2.0));
	double tmp;
	if (phi2 <= -1.6e-18) {
		tmp = t_2;
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi1))
	t_1 = Float64((fma(sin(Float64(phi2 * -0.5)), cos(Float64(0.5 * phi1)), Float64(t_0 * cos(Float64(phi2 * -0.5)))) ^ 2.0) + Float64(cos(phi1) * Float64(cos(phi2) * fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5))))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	t_3 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (t_0 ^ 2.0))
	tmp = 0.0
	if (phi2 <= -1.6e-18)
		tmp = t_2;
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(t$95$0 * N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.6e-18], t$95$2, If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.6e-18 or 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\phi_2}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \color{blue}{\frac{-1}{2}}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{-1}{2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       17. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       19. lift-sin.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \color{blue}{\sin \left(\phi_1 \cdot 0.5\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

       20. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \color{blue}{\left(0.5 \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

       23. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right) + \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right)} + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{\phi_2}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \color{blue}{\frac{-1}{2}}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{-1}{2}\right)}, \cos \left(\phi_1 \cdot \frac{1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       17. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       19. lift-sin.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \color{blue}{\sin \left(\phi_1 \cdot 0.5\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

       20. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       22. lower-*.f64 76.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \color{blue}{\left(0.5 \cdot \phi_1\right)} \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

       23. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\frac{\phi_2}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       24. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot \frac{-1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{-2}\right)}\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 76.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot -0.5\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot -0.5\right)\right)\right)}}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

   if -1.6e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := \sin \left(-0.5 \cdot \phi_2\right)\\ t_2 := \sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\\ t_3 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {t\_0}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_2}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), t\_0, \cos \left(0.5 \cdot \phi_1\right) \cdot t\_1\right)\right)}^{2}\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_2}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(t\_1, \cos \left(-0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot t\_0\right)\right)}^{2}\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1 (sin (* -0.5 phi2)))
        (t_2
         (sqrt
          (+
           (pow
            (fma
             (sin (* phi1 0.5))
             (cos (/ phi2 -2.0))
             (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
            2.0)
           (*
            (cos phi1)
            (* (cos phi2) (fma -0.5 (cos (- lambda2 lambda1)) 0.5))))))
        (t_3
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow t_0 2.0))))
   (if (<= phi2 -1.6e-18)
     (*
      R
      (*
       2.0
       (atan2
        t_2
        (sqrt
         (-
          1.0
          (fma
           (cos phi1)
           (* (cos phi2) (+ 0.5 (* -0.5 (cos lambda2))))
           (pow
            (fma (cos (* -0.5 phi2)) t_0 (* (cos (* 0.5 phi1)) t_1))
            2.0)))))))
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (*
        R
        (*
         2.0
         (atan2
          t_2
          (sqrt
           (-
            1.0
            (fma
             (cos phi1)
             (* (cos phi2) (+ 0.5 (* -0.5 (cos (- lambda1)))))
             (pow
              (fma t_1 (cos (* -0.5 phi1)) (* (cos (* phi2 0.5)) t_0))
              2.0)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1));
	double t_1 = sin((-0.5 * phi2));
	double t_2 = sqrt((pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + (cos(phi1) * (cos(phi2) * fma(-0.5, cos((lambda2 - lambda1)), 0.5)))));
	double t_3 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(t_0, 2.0));
	double tmp;
	if (phi2 <= -1.6e-18) {
		tmp = R * (2.0 * atan2(t_2, sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 + (-0.5 * cos(lambda2)))), pow(fma(cos((-0.5 * phi2)), t_0, (cos((0.5 * phi1)) * t_1)), 2.0))))));
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(t_2, sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 + (-0.5 * cos(-lambda1)))), pow(fma(t_1, cos((-0.5 * phi1)), (cos((phi2 * 0.5)) * t_0)), 2.0))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi1))
	t_1 = sin(Float64(-0.5 * phi2))
	t_2 = sqrt(Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + Float64(cos(phi1) * Float64(cos(phi2) * fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)))))
	t_3 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (t_0 ^ 2.0))
	tmp = 0.0
	if (phi2 <= -1.6e-18)
		tmp = Float64(R * Float64(2.0 * atan(t_2, sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 + Float64(-0.5 * cos(lambda2)))), (fma(cos(Float64(-0.5 * phi2)), t_0, Float64(cos(Float64(0.5 * phi1)) * t_1)) ^ 2.0)))))));
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_2, sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 + Float64(-0.5 * cos(Float64(-lambda1))))), (fma(t_1, cos(Float64(-0.5 * phi1)), Float64(cos(Float64(phi2 * 0.5)) * t_0)) ^ 2.0)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.6e-18], N[(R * N[(2.0 * N[ArcTan[t$95$2 / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$0 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[t$95$2 / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[(-lambda1)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(t$95$1 * N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.6e-18

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   12. Applied rewrites 57.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   if -1.6e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   12. Applied rewrites 57.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]
   13. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right) + \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       3. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right) + \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       6. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\phi_2 \cdot \frac{1}{-2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       8. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       10. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       13. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       14. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       15. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \cos \left(\phi_2 \cdot \frac{1}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       16. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{\phi_2}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       17. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{\phi_2}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{\phi_2}{-2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2}\right)}}\right) \]

       19. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{\phi_2}{-2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)}^{2}\right)}}\right) \]

       20. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{\phi_2}{-2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)}^{2}\right)}}\right) \]

       21. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\sin \left(\frac{\phi_2}{-2}\right) \cdot \cos \left(\phi_1 \cdot \frac{1}{2}\right) + \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{-2}\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 57.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(-0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}^{2}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := \sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\\ t_2 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), t\_0, \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\\ t_3 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {t\_0}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_1}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), t\_2\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_1}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right) \cdot \cos \phi_2, t\_2\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1
         (sqrt
          (+
           (pow
            (fma
             (sin (* phi1 0.5))
             (cos (/ phi2 -2.0))
             (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
            2.0)
           (*
            (cos phi1)
            (* (cos phi2) (fma -0.5 (cos (- lambda2 lambda1)) 0.5))))))
        (t_2
         (pow
          (fma
           (cos (* -0.5 phi2))
           t_0
           (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
          2.0))
        (t_3
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow t_0 2.0))))
   (if (<= phi2 -1.6e-18)
     (*
      R
      (*
       2.0
       (atan2
        t_1
        (sqrt
         (-
          1.0
          (fma
           (cos phi1)
           (* (cos phi2) (+ 0.5 (* -0.5 (cos lambda2))))
           t_2))))))
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (*
        R
        (*
         2.0
         (atan2
          t_1
          (sqrt
           (-
            1.0
            (fma
             (cos phi1)
             (* (fma (cos lambda1) -0.5 0.5) (cos phi2))
             t_2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1));
	double t_1 = sqrt((pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + (cos(phi1) * (cos(phi2) * fma(-0.5, cos((lambda2 - lambda1)), 0.5)))));
	double t_2 = pow(fma(cos((-0.5 * phi2)), t_0, (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0);
	double t_3 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(t_0, 2.0));
	double tmp;
	if (phi2 <= -1.6e-18) {
		tmp = R * (2.0 * atan2(t_1, sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 + (-0.5 * cos(lambda2)))), t_2)))));
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(t_1, sqrt((1.0 - fma(cos(phi1), (fma(cos(lambda1), -0.5, 0.5) * cos(phi2)), t_2)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi1))
	t_1 = sqrt(Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + Float64(cos(phi1) * Float64(cos(phi2) * fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)))))
	t_2 = fma(cos(Float64(-0.5 * phi2)), t_0, Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0
	t_3 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (t_0 ^ 2.0))
	tmp = 0.0
	if (phi2 <= -1.6e-18)
		tmp = Float64(R * Float64(2.0 * atan(t_1, sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 + Float64(-0.5 * cos(lambda2)))), t_2))))));
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_1, sqrt(Float64(1.0 - fma(cos(phi1), Float64(fma(cos(lambda1), -0.5, 0.5) * cos(phi2)), t_2))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$0 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.6e-18], N[(R * N[(2.0 * N[ArcTan[t$95$1 / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[t$95$1 / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.6e-18

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   12. Applied rewrites 57.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   if -1.6e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   12. Applied rewrites 57.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right)}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right) \cdot \color{blue}{\cos \phi_2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 57.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right) \cdot \color{blue}{\cos \phi_2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       4. lift-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right) \cdot \cos \color{blue}{\phi_2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       5. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{-1}{2} \cdot \cos \left(-\lambda_1\right) + \frac{1}{2}\right) \cdot \cos \color{blue}{\phi_2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{-1}{2} \cdot \cos \left(-\lambda_1\right) + \frac{1}{2}\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\cos \left(-\lambda_1\right) \cdot \frac{-1}{2} + \frac{1}{2}\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       8. lower-fma.f64 57.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(-\lambda_1\right), -0.5, 0.5\right) \cdot \cos \color{blue}{\phi_2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       9. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(-\lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       10. lift-neg.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right), \frac{-1}{2}, \frac{1}{2}\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       11. cos-neg N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       12. lower-cos.f64 57.3

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 57.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right) \cdot \color{blue}{\cos \phi_2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), t\_1, \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\\ t_3 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {t\_1}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), t\_2\right)}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right) \cdot \cos \phi_2, t\_2\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          (pow
           (fma
            (sin (* phi1 0.5))
            (cos (/ phi2 -2.0))
            (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
           2.0)
          (*
           (cos phi1)
           (* (cos phi2) (fma -0.5 (cos (- lambda2 lambda1)) 0.5)))))
        (t_1 (sin (* 0.5 phi1)))
        (t_2
         (pow
          (fma
           (cos (* -0.5 phi2))
           t_1
           (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
          2.0))
        (t_3
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow t_1 2.0))))
   (if (<= phi2 -1.6e-18)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (fma (cos phi1) (* (cos phi2) (+ 0.5 (* -0.5 (cos lambda2)))) t_2))
        (sqrt (- 1.0 t_0)))))
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt t_0)
          (sqrt
           (-
            1.0
            (fma
             (cos phi1)
             (* (fma (cos lambda1) -0.5 0.5) (cos phi2))
             t_2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + (cos(phi1) * (cos(phi2) * fma(-0.5, cos((lambda2 - lambda1)), 0.5)));
	double t_1 = sin((0.5 * phi1));
	double t_2 = pow(fma(cos((-0.5 * phi2)), t_1, (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0);
	double t_3 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(t_1, 2.0));
	double tmp;
	if (phi2 <= -1.6e-18) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (0.5 + (-0.5 * cos(lambda2)))), t_2)), sqrt((1.0 - t_0))));
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - fma(cos(phi1), (fma(cos(lambda1), -0.5, 0.5) * cos(phi2)), t_2)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + Float64(cos(phi1) * Float64(cos(phi2) * fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5))))
	t_1 = sin(Float64(0.5 * phi1))
	t_2 = fma(cos(Float64(-0.5 * phi2)), t_1, Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0
	t_3 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (t_1 ^ 2.0))
	tmp = 0.0
	if (phi2 <= -1.6e-18)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 + Float64(-0.5 * cos(lambda2)))), t_2)), sqrt(Float64(1.0 - t_0)))));
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - fma(cos(phi1), Float64(fma(cos(lambda1), -0.5, 0.5) * cos(phi2)), t_2))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$1 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.6e-18], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.6e-18

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

   if -1.6e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   12. Applied rewrites 57.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right)}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right) \cdot \color{blue}{\cos \phi_2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 57.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right) \cdot \color{blue}{\cos \phi_2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       4. lift-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right) \cdot \cos \color{blue}{\phi_2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       5. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{-1}{2} \cdot \cos \left(-\lambda_1\right) + \frac{1}{2}\right) \cdot \cos \color{blue}{\phi_2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{-1}{2} \cdot \cos \left(-\lambda_1\right) + \frac{1}{2}\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\cos \left(-\lambda_1\right) \cdot \frac{-1}{2} + \frac{1}{2}\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       8. lower-fma.f64 57.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(-\lambda_1\right), -0.5, 0.5\right) \cdot \cos \color{blue}{\phi_2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       9. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(-\lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       10. lift-neg.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right), \frac{-1}{2}, \frac{1}{2}\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       11. cos-neg N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       12. lower-cos.f64 57.3

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 57.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right) \cdot \color{blue}{\cos \phi_2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 77.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), t\_0, \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\\ t_2 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {t\_0}^{2}\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), t\_1\right)\\ \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right) \cdot \cos \phi_2, t\_1\right)}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1
         (pow
          (fma
           (cos (* -0.5 phi2))
           t_0
           (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
          2.0))
        (t_2
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow t_0 2.0)))
        (t_3
         (fma (cos phi1) (* (cos phi2) (+ 0.5 (* -0.5 (cos lambda2)))) t_1)))
   (if (<= phi2 -1.6e-18)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (+
            (pow
             (fma
              (sin (* phi1 0.5))
              (cos (/ phi2 -2.0))
              (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
             2.0)
            (*
             (cos phi1)
             (* (cos phi2) (fma -0.5 (cos (- lambda2 lambda1)) 0.5)))))
          (sqrt
           (-
            1.0
            (fma
             (cos phi1)
             (* (fma (cos lambda1) -0.5 0.5) (cos phi2))
             t_1))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1));
	double t_1 = pow(fma(cos((-0.5 * phi2)), t_0, (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0);
	double t_2 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(t_0, 2.0));
	double t_3 = fma(cos(phi1), (cos(phi2) * (0.5 + (-0.5 * cos(lambda2)))), t_1);
	double tmp;
	if (phi2 <= -1.6e-18) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + (cos(phi1) * (cos(phi2) * fma(-0.5, cos((lambda2 - lambda1)), 0.5))))), sqrt((1.0 - fma(cos(phi1), (fma(cos(lambda1), -0.5, 0.5) * cos(phi2)), t_1)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi1))
	t_1 = fma(cos(Float64(-0.5 * phi2)), t_0, Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0
	t_2 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (t_0 ^ 2.0))
	t_3 = fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 + Float64(-0.5 * cos(lambda2)))), t_1)
	tmp = 0.0
	if (phi2 <= -1.6e-18)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + Float64(cos(phi1) * Float64(cos(phi2) * fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5))))), sqrt(Float64(1.0 - fma(cos(phi1), Float64(fma(cos(lambda1), -0.5, 0.5) * cos(phi2)), t_1))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$0 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[phi2, -1.6e-18], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.6e-18

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \lambda_2}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \lambda_2}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \lambda_2}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \lambda_2}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 55.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   if -1.6e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   12. Applied rewrites 57.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right)}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right) \cdot \color{blue}{\cos \phi_2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 57.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right) \cdot \color{blue}{\cos \phi_2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       4. lift-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right) \cdot \cos \color{blue}{\phi_2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       5. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{-1}{2} \cdot \cos \left(-\lambda_1\right) + \frac{1}{2}\right) \cdot \cos \color{blue}{\phi_2}, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{-1}{2} \cdot \cos \left(-\lambda_1\right) + \frac{1}{2}\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       7. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\cos \left(-\lambda_1\right) \cdot \frac{-1}{2} + \frac{1}{2}\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       8. lower-fma.f64 57.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(-\lambda_1\right), -0.5, 0.5\right) \cdot \cos \color{blue}{\phi_2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       9. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(-\lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       10. lift-neg.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right), \frac{-1}{2}, \frac{1}{2}\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       11. cos-neg N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \lambda_1, \frac{-1}{2}, \frac{1}{2}\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

       12. lower-cos.f64 57.3

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right) \cdot \cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 57.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \lambda_1, -0.5, 0.5\right) \cdot \color{blue}{\cos \phi_2}, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), t\_0, \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\\ t_2 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right), t\_1\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {t\_0}^{2}\right)\\ t_4 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), t\_1\right)\\ \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1
         (pow
          (fma
           (cos (* -0.5 phi2))
           t_0
           (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
          2.0))
        (t_2
         (fma
          (cos phi1)
          (* (cos phi2) (+ 0.5 (* -0.5 (cos (- lambda1)))))
          t_1))
        (t_3
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow t_0 2.0)))
        (t_4
         (fma (cos phi1) (* (cos phi2) (+ 0.5 (* -0.5 (cos lambda2)))) t_1)))
   (if (<= phi2 -1.6e-18)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1));
	double t_1 = pow(fma(cos((-0.5 * phi2)), t_0, (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0);
	double t_2 = fma(cos(phi1), (cos(phi2) * (0.5 + (-0.5 * cos(-lambda1)))), t_1);
	double t_3 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(t_0, 2.0));
	double t_4 = fma(cos(phi1), (cos(phi2) * (0.5 + (-0.5 * cos(lambda2)))), t_1);
	double tmp;
	if (phi2 <= -1.6e-18) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi1))
	t_1 = fma(cos(Float64(-0.5 * phi2)), t_0, Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0
	t_2 = fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 + Float64(-0.5 * cos(Float64(-lambda1))))), t_1)
	t_3 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (t_0 ^ 2.0))
	t_4 = fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 + Float64(-0.5 * cos(lambda2)))), t_1)
	tmp = 0.0
	if (phi2 <= -1.6e-18)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$0 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[(-lambda1)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[phi2, -1.6e-18], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.6e-18

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \lambda_2}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \lambda_2}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \lambda_2}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \lambda_2}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 55.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   if -1.6e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right)}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right)}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(-\lambda_1\right)\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 55.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 76.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {t\_0}^{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), t\_0, \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow t_0 2.0)))
        (t_2
         (fma
          (cos phi1)
          (* (cos phi2) (+ 0.5 (* -0.5 (cos lambda2))))
          (pow
           (fma
            (cos (* -0.5 phi2))
            t_0
            (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
           2.0)))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -1.6e-18)
     t_3
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1));
	double t_1 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(t_0, 2.0));
	double t_2 = fma(cos(phi1), (cos(phi2) * (0.5 + (-0.5 * cos(lambda2)))), pow(fma(cos((-0.5 * phi2)), t_0, (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -1.6e-18) {
		tmp = t_3;
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi1))
	t_1 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (t_0 ^ 2.0))
	t_2 = fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 + Float64(-0.5 * cos(lambda2)))), (fma(cos(Float64(-0.5 * phi2)), t_0, Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -1.6e-18)
		tmp = t_3;
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$0 + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.6e-18], t$95$3, If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.6e-18 or 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \lambda_2}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \lambda_2}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       7. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \lambda_2}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \lambda_2}\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \lambda_2\right), {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 55.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_2\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   if -1.6e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 76.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}\\ t_1 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (-
           (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
           (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
          2.0))
        (t_1
         (+
          (pow
           (fma
            (sin (* phi1 0.5))
            (cos (/ phi2 -2.0))
            (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
           2.0)
          (* (cos phi2) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))))
        (t_2 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
        (t_3 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0))))
   (if (<= phi2 -1.6e-18)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0);
	double t_1 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + (cos(phi2) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double t_2 = fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0));
	double t_3 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double tmp;
	if (phi2 <= -1.6e-18) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0
	t_1 = Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + Float64(cos(phi2) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	t_2 = fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_3 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	tmp = 0.0
	if (phi2 <= -1.6e-18)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.6e-18], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.6e-18

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -1.6e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 59.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 59.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

       4. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       6. lower--.f64 55.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 55.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (-
           (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
           (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
          2.0))
        (t_1 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
        (t_2 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -1.6e-18)
     t_3
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0);
	double t_1 = fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0));
	double t_2 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -1.6e-18) {
		tmp = t_3;
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0
	t_1 = fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_2 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -1.6e-18)
		tmp = t_3;
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.6e-18], t$95$3, If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.6e-18 or 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -1.6e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0}}{\sqrt{\frac{2 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right), \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) \cdot 2\right)}{2}}}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -1.55 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (*
          R
          (*
           2.0
           (atan2
            (sqrt
             (+
              (pow
               (fma
                (sin (* phi1 0.5))
                (cos (/ phi2 -2.0))
                (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
               2.0)
              (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
            (sqrt
             (/
              (-
               2.0
               (fma
                (fma -0.5 (cos (- lambda2 lambda1)) 0.5)
                (+ (cos (+ phi2 phi1)) (cos (- phi2 phi1)))
                (* (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5))))) 2.0)))
              2.0))))))
        (t_2
         (fma
          (cos phi1)
          (pow
           (-
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1)))
            (* (cos (* 0.5 lambda1)) (sin (* 0.5 lambda2))))
           2.0)
          (pow (sin (* 0.5 phi1)) 2.0))))
   (if (<= phi2 -1.55e-18)
     t_1
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = R * (2.0 * atan2(sqrt((pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(((2.0 - fma(fma(-0.5, cos((lambda2 - lambda1)), 0.5), (cos((phi2 + phi1)) + cos((phi2 - phi1))), ((0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))))) * 2.0))) / 2.0))));
	double t_2 = fma(cos(phi1), pow(((cos((0.5 * lambda2)) * sin((0.5 * lambda1))) - (cos((0.5 * lambda1)) * sin((0.5 * lambda2)))), 2.0), pow(sin((0.5 * phi1)), 2.0));
	double tmp;
	if (phi2 <= -1.55e-18) {
		tmp = t_1;
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(R * Float64(2.0 * atan(sqrt(Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(Float64(Float64(2.0 - fma(fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5), Float64(cos(Float64(phi2 + phi1)) + cos(Float64(phi2 - phi1))), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5))))) * 2.0))) / 2.0)))))
	t_2 = fma(cos(phi1), (Float64(Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1))) - Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(0.5 * lambda2)))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))
	tmp = 0.0
	if (phi2 <= -1.55e-18)
		tmp = t_1;
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(2.0 - N[(N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.55e-18], t$95$1, If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.55000000000000003e-18 or 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 63.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\frac{2 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right), \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) \cdot 2\right)}{2}}}}\right) \]

   if -1.55000000000000003e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 77.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. lower-*.f64 79.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       24. lower-*.f64 78.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       9. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       11. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{\lambda_2}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       13. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       15. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)} - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       17. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       18. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       19. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       20. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\lambda_2}{2}\right)\right)\right)}}\right) \]

       21. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{2}\right)}\right)\right)}}\right) \]

       22. mult-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}\right) \]

       24. lower-*.f64 98.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\lambda_2 \cdot 0.5\right)}\right)\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\lambda_2 \cdot \frac{1}{2}\right) - \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(\lambda_1 \cdot 0.5\right) \cdot \cos \left(\lambda_2 \cdot 0.5\right) - \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot 0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) - \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right) - \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(0.5 \cdot \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 65.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1}}{\sqrt{\frac{2 - \mathsf{fma}\left(t\_2, \cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right), \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) \cdot 2\right)}{2}}}\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-48}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-27}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t\_2\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (sin (* phi1 0.5))
           (cos (/ phi2 -2.0))
           (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
          2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (fma -0.5 (cos (- lambda2 lambda1)) 0.5))
        (t_3
         (*
          R
          (*
           2.0
           (atan2
            (sqrt (+ t_0 (* (* (* (cos phi1) (cos phi2)) t_1) t_1)))
            (sqrt
             (/
              (-
               2.0
               (fma
                t_2
                (+ (cos (+ phi2 phi1)) (cos (- phi2 phi1)))
                (* (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5))))) 2.0)))
              2.0)))))))
   (if (<= t_1 -2e-48)
     t_3
     (if (<= t_1 5e-27)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ t_0 (* (cos phi1) (* (cos phi2) t_2))))
          (sqrt
           (-
            1.0
            (pow
             (fma
              (cos (* -0.5 phi2))
              (sin (* 0.5 phi1))
              (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
             2.0))))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = fma(-0.5, cos((lambda2 - lambda1)), 0.5);
	double t_3 = R * (2.0 * atan2(sqrt((t_0 + (((cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt(((2.0 - fma(t_2, (cos((phi2 + phi1)) + cos((phi2 - phi1))), ((0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))))) * 2.0))) / 2.0))));
	double tmp;
	if (t_1 <= -2e-48) {
		tmp = t_3;
	} else if (t_1 <= 5e-27) {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (cos(phi1) * (cos(phi2) * t_2)))), sqrt((1.0 - pow(fma(cos((-0.5 * phi2)), sin((0.5 * phi1)), (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt(Float64(Float64(2.0 - fma(t_2, Float64(cos(Float64(phi2 + phi1)) + cos(Float64(phi2 - phi1))), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5))))) * 2.0))) / 2.0)))))
	tmp = 0.0
	if (t_1 <= -2e-48)
		tmp = t_3;
	elseif (t_1 <= 5e-27)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * t_2)))), sqrt(Float64(1.0 - (fma(cos(Float64(-0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0))))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(2.0 - N[(t$95$2 * N[(N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-48], t$95$3, If[LessEqual[t$95$1, 5e-27], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -1.9999999999999999e-48 or 5.0000000000000002e-27 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 63.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\frac{2 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right), \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) \cdot 2\right)}{2}}}}\right) \]

   if -1.9999999999999999e-48 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 5.0000000000000002e-27

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   12. Applied rewrites 57.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   14. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       11. lower-*.f64 38.5

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

   15. Applied rewrites 38.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 65.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2}\\ t_1 := \frac{\mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \left(\frac{\left(\phi_2 - \phi_1\right) - \left(\phi_2 - \phi_1\right)}{-2}\right) - \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_4 := t\_0 + \cos \phi_1 \cdot \left(0.5 + -0.5 \cdot t\_3\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-48}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, t\_3, 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (sin (* phi1 0.5))
           (cos (/ phi2 -2.0))
           (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
          2.0))
        (t_1
         (/
          (fma
           (+ (cos (- phi2 phi1)) (cos (+ phi2 phi1)))
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
           (-
            (cos (/ (- (- phi2 phi1) (- phi2 phi1)) -2.0))
            (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
          2.0))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (cos (- lambda2 lambda1)))
        (t_4 (+ t_0 (* (cos phi1) (+ 0.5 (* -0.5 t_3))))))
   (if (<= t_2 -2e-48)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_0 (* (* (* (cos phi1) (cos phi2)) t_2) t_2)))
        (sqrt
         (-
          1.0
          (fma
           (fma -0.5 t_3 0.5)
           (* (cos phi2) (cos phi1))
           (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5)))))))))))
     (if (<= t_2 2e-5)
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0);
	double t_1 = fma((cos((phi2 - phi1)) + cos((phi2 + phi1))), (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))), (cos((((phi2 - phi1) - (phi2 - phi1)) / -2.0)) - cos((2.0 * (0.5 * (phi1 - phi2)))))) / 2.0;
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = cos((lambda2 - lambda1));
	double t_4 = t_0 + (cos(phi1) * (0.5 + (-0.5 * t_3)));
	double tmp;
	if (t_2 <= -2e-48) {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (((cos(phi1) * cos(phi2)) * t_2) * t_2))), sqrt((1.0 - fma(fma(-0.5, t_3, 0.5), (cos(phi2) * cos(phi1)), (0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))))))))));
	} else if (t_2 <= 2e-5) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0
	t_1 = Float64(fma(Float64(cos(Float64(phi2 - phi1)) + cos(Float64(phi2 + phi1))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))), Float64(cos(Float64(Float64(Float64(phi2 - phi1) - Float64(phi2 - phi1)) / -2.0)) - cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))) / 2.0)
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = cos(Float64(lambda2 - lambda1))
	t_4 = Float64(t_0 + Float64(cos(phi1) * Float64(0.5 + Float64(-0.5 * t_3))))
	tmp = 0.0
	if (t_2 <= -2e-48)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2))), sqrt(Float64(1.0 - fma(fma(-0.5, t_3, 0.5), Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5)))))))))));
	elseif (t_2 <= 2e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(N[(N[(phi2 - phi1), $MachinePrecision] - N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]], $MachinePrecision] - N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 + N[(-0.5 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-48], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(-0.5 * t$95$3 + 0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -1.9999999999999999e-48

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}}\right) \]

   if -1.9999999999999999e-48 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 2.00000000000000016e-5

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lower--.f64 56.2

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right)}}\right) \]
   14. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\frac{1}{2}} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)}}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\frac{1}{2} + \color{blue}{\frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)}\right)\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right)\right)}}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)}}\right) \]

       6. lower--.f64 52.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)}}\right) \]

   15. Applied rewrites 52.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right)}}\right) \]

   if 2.00000000000000016e-5 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 57.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\frac{\mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \left(\frac{\left(\phi_2 - \phi_1\right) - \left(\phi_2 - \phi_1\right)}{-2}\right) - \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 58.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{\mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \left(\frac{\left(\phi_2 - \phi_1\right) - \left(\phi_2 - \phi_1\right)}{-2}\right) - \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \color{blue}{\frac{\mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \left(\frac{\left(\phi_2 - \phi_1\right) - \left(\phi_2 - \phi_1\right)}{-2}\right) - \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}{2}}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 65.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\\ t_3 := \frac{\mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \left(\frac{\left(\phi_2 - \phi_1\right) - \left(\phi_2 - \phi_1\right)}{-2}\right) - \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}{2}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-48}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1}}{\sqrt{1 - \mathsf{fma}\left(t\_2, \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t\_2\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (sin (* phi1 0.5))
           (cos (/ phi2 -2.0))
           (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
          2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (fma -0.5 (cos (- lambda2 lambda1)) 0.5))
        (t_3
         (/
          (fma
           (+ (cos (- phi2 phi1)) (cos (+ phi2 phi1)))
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
           (-
            (cos (/ (- (- phi2 phi1) (- phi2 phi1)) -2.0))
            (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
          2.0)))
   (if (<= t_1 -2e-48)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_0 (* (* (* (cos phi1) (cos phi2)) t_1) t_1)))
        (sqrt
         (-
          1.0
          (fma
           t_2
           (* (cos phi2) (cos phi1))
           (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5)))))))))))
     (if (<= t_1 2e-5)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ t_0 (* (cos phi1) (* (cos phi2) t_2))))
          (sqrt
           (-
            1.0
            (pow
             (fma
              (cos (* -0.5 phi2))
              (sin (* 0.5 phi1))
              (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
             2.0))))))
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = fma(-0.5, cos((lambda2 - lambda1)), 0.5);
	double t_3 = fma((cos((phi2 - phi1)) + cos((phi2 + phi1))), (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))), (cos((((phi2 - phi1) - (phi2 - phi1)) / -2.0)) - cos((2.0 * (0.5 * (phi1 - phi2)))))) / 2.0;
	double tmp;
	if (t_1 <= -2e-48) {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (((cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt((1.0 - fma(t_2, (cos(phi2) * cos(phi1)), (0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))))))))));
	} else if (t_1 <= 2e-5) {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (cos(phi1) * (cos(phi2) * t_2)))), sqrt((1.0 - pow(fma(cos((-0.5 * phi2)), sin((0.5 * phi1)), (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)
	t_3 = Float64(fma(Float64(cos(Float64(phi2 - phi1)) + cos(Float64(phi2 + phi1))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))), Float64(cos(Float64(Float64(Float64(phi2 - phi1) - Float64(phi2 - phi1)) / -2.0)) - cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))) / 2.0)
	tmp = 0.0
	if (t_1 <= -2e-48)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt(Float64(1.0 - fma(t_2, Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5)))))))))));
	elseif (t_1 <= 2e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * t_2)))), sqrt(Float64(1.0 - (fma(cos(Float64(-0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(N[(N[(phi2 - phi1), $MachinePrecision] - N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]], $MachinePrecision] - N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-48], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$2 * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -1.9999999999999999e-48

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}}\right) \]

   if -1.9999999999999999e-48 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 2.00000000000000016e-5

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   12. Applied rewrites 57.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   14. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       11. lower-*.f64 38.5

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

   15. Applied rewrites 38.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}\right) \]

   if 2.00000000000000016e-5 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 57.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\frac{\mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \left(\frac{\left(\phi_2 - \phi_1\right) - \left(\phi_2 - \phi_1\right)}{-2}\right) - \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 58.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{\mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \left(\frac{\left(\phi_2 - \phi_1\right) - \left(\phi_2 - \phi_1\right)}{-2}\right) - \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \color{blue}{\frac{\mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \left(\frac{\left(\phi_2 - \phi_1\right) - \left(\phi_2 - \phi_1\right)}{-2}\right) - \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}{2}}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 65.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_1 := \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \frac{\mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right), t\_0, \cos \left(\frac{\left(\phi_2 - \phi_1\right) - \left(\phi_2 - \phi_1\right)}{-2}\right) - t\_1\right)}{2}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-48}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2}}{\sqrt{\left(1 - t\_0 \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right) - \left(0.5 - 0.5 \cdot t\_1\right)}}\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
        (t_1 (cos (* 2.0 (* 0.5 (- phi1 phi2)))))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3
         (/
          (fma
           (+ (cos (- phi2 phi1)) (cos (+ phi2 phi1)))
           t_0
           (- (cos (/ (- (- phi2 phi1) (- phi2 phi1)) -2.0)) t_1))
          2.0)))
   (if (<= t_2 -2e-48)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_2) t_2)))
        (sqrt
         (- (- 1.0 (* t_0 (* (cos phi2) (cos phi1)))) (- 0.5 (* 0.5 t_1)))))))
     (if (<= t_2 2e-5)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (+
            (pow
             (fma
              (sin (* phi1 0.5))
              (cos (/ phi2 -2.0))
              (* (cos (* phi1 0.5)) (sin (/ phi2 -2.0))))
             2.0)
            (*
             (cos phi1)
             (* (cos phi2) (fma -0.5 (cos (- lambda2 lambda1)) 0.5)))))
          (sqrt
           (-
            1.0
            (pow
             (fma
              (cos (* -0.5 phi2))
              (sin (* 0.5 phi1))
              (* (cos (* 0.5 phi1)) (sin (* -0.5 phi2))))
             2.0))))))
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))));
	double t_1 = cos((2.0 * (0.5 * (phi1 - phi2))));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = fma((cos((phi2 - phi1)) + cos((phi2 + phi1))), t_0, (cos((((phi2 - phi1) - (phi2 - phi1)) / -2.0)) - t_1)) / 2.0;
	double tmp;
	if (t_2 <= -2e-48) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_2) * t_2))), sqrt(((1.0 - (t_0 * (cos(phi2) * cos(phi1)))) - (0.5 - (0.5 * t_1))))));
	} else if (t_2 <= 2e-5) {
		tmp = R * (2.0 * atan2(sqrt((pow(fma(sin((phi1 * 0.5)), cos((phi2 / -2.0)), (cos((phi1 * 0.5)) * sin((phi2 / -2.0)))), 2.0) + (cos(phi1) * (cos(phi2) * fma(-0.5, cos((lambda2 - lambda1)), 0.5))))), sqrt((1.0 - pow(fma(cos((-0.5 * phi2)), sin((0.5 * phi1)), (cos((0.5 * phi1)) * sin((-0.5 * phi2)))), 2.0)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))
	t_1 = cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2))))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(fma(Float64(cos(Float64(phi2 - phi1)) + cos(Float64(phi2 + phi1))), t_0, Float64(cos(Float64(Float64(Float64(phi2 - phi1) - Float64(phi2 - phi1)) / -2.0)) - t_1)) / 2.0)
	tmp = 0.0
	if (t_2 <= -2e-48)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2))), sqrt(Float64(Float64(1.0 - Float64(t_0 * Float64(cos(phi2) * cos(phi1)))) - Float64(0.5 - Float64(0.5 * t_1)))))));
	elseif (t_2 <= 2e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(phi2 / -2.0)), Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(phi2 / -2.0)))) ^ 2.0) + Float64(cos(phi1) * Float64(cos(phi2) * fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5))))), sqrt(Float64(1.0 - (fma(cos(Float64(-0.5 * phi2)), sin(Float64(0.5 * phi1)), Float64(cos(Float64(0.5 * phi1)) * sin(Float64(-0.5 * phi2)))) ^ 2.0))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[Cos[N[(N[(N[(phi2 - phi1), $MachinePrecision] - N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision] / -2.0), $MachinePrecision]], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-48], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[(t$95$0 * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 - N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -1.9999999999999999e-48

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}\right) \]

   if -1.9999999999999999e-48 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 2.00000000000000016e-5

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 75.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

      12. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

      13. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

      14. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}\right)}}\right) \]

      15. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      16. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

      17. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)\right)}}\right) \]

   9. Applied rewrites 76.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} + \frac{-1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   12. Applied rewrites 57.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(-\lambda_1\right)\right), {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   14. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\frac{-1}{2}, \cos \left(\lambda_2 - \lambda_1\right), \frac{1}{2}\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right), \sin \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

       11. lower-*.f64 38.5

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}}}\right) \]

   15. Applied rewrites 38.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\right)}}{\sqrt{1 - {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)\right)\right)}^{\color{blue}{2}}}}\right) \]

   if 2.00000000000000016e-5 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 57.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\frac{\mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \left(\frac{\left(\phi_2 - \phi_1\right) - \left(\phi_2 - \phi_1\right)}{-2}\right) - \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 58.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{\mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \left(\frac{\left(\phi_2 - \phi_1\right) - \left(\phi_2 - \phi_1\right)}{-2}\right) - \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \color{blue}{\frac{\mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \left(\frac{\left(\phi_2 - \phi_1\right) - \left(\phi_2 - \phi_1\right)}{-2}\right) - \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}{2}}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 62.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot t\_0\\ t_2 := {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\\ t_3 := \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), t\_2\right)\\ t_4 := \mathsf{fma}\left(\cos \phi_2, t\_0, t\_2\right)\\ \mathbf{if}\;\phi_2 \leq -1.85:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{elif}\;\phi_2 \leq 11500000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi1) t_0)))
        (t_2 (pow (sin (* -0.5 phi2)) 2.0))
        (t_3 (fma (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))) t_2))
        (t_4 (fma (cos phi2) t_0 t_2)))
   (if (<= phi2 -1.85)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (if (<= phi2 11500000.0)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * t_0);
	double t_2 = pow(sin((-0.5 * phi2)), 2.0);
	double t_3 = fma(cos(phi2), (0.5 - (0.5 * cos((lambda1 - lambda2)))), t_2);
	double t_4 = fma(cos(phi2), t_0, t_2);
	double tmp;
	if (phi2 <= -1.85) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else if (phi2 <= 11500000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * t_0))
	t_2 = sin(Float64(-0.5 * phi2)) ^ 2.0
	t_3 = fma(cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), t_2)
	t_4 = fma(cos(phi2), t_0, t_2)
	tmp = 0.0
	if (phi2 <= -1.85)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	elseif (phi2 <= 11500000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + t$95$2), $MachinePrecision]}, If[LessEqual[phi2, -1.85], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 11500000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.8500000000000001

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.5

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -1.8500000000000001 < phi2 < 1.15e7

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 52.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 52.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 50.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 50.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 1.15e7 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

   5. Applied rewrites 60.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      3. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      9. lower-*.f64 44.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   8. Applied rewrites 44.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 44.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 44.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 62.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_2 := \frac{1}{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right), \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) \cdot 2\right)}}\\ t_3 := t\_1 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;t\_1 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0 \leq 0.01:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_2
         (/
          1.0
          (/
           2.0
           (fma
            (fma -0.5 (cos (- lambda2 lambda1)) 0.5)
            (+ (cos (+ phi2 phi1)) (cos (- phi2 phi1)))
            (* (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5))))) 2.0)))))
        (t_3
         (+ t_1 (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))))
   (if (<= (+ t_1 (* (* (* (cos phi1) (cos phi2)) t_0) t_0)) 0.01)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_2 = 1.0 / (2.0 / fma(fma(-0.5, cos((lambda2 - lambda1)), 0.5), (cos((phi2 + phi1)) + cos((phi2 - phi1))), ((0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))))) * 2.0)));
	double t_3 = t_1 + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double tmp;
	if ((t_1 + (((cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 0.01) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_2 = Float64(1.0 / Float64(2.0 / fma(fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5), Float64(cos(Float64(phi2 + phi1)) + cos(Float64(phi2 - phi1))), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5))))) * 2.0))))
	t_3 = Float64(t_1 + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	tmp = 0.0
	if (Float64(t_1 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 0.01)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(2.0 / N[(N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 0.01], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 52.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 52.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 50.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 50.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 0.0100000000000000002 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 58.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\frac{1}{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right), \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) \cdot 2\right)}}}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 58.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right), \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) \cdot 2\right)}}}}{\sqrt{1 - \color{blue}{\frac{1}{\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right), \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) \cdot 2\right)}}}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 62.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0}}{\sqrt{\left|\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right|}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
        (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
      (sqrt
       (fabs
        (-
         (+ 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
         (*
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
          (* (cos phi2) (cos phi1)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(fabs(((0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1))))))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    code = r * (2.0d0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(abs(((0.5d0 + (0.5d0 * cos((2.0d0 * (0.5d0 * (phi1 - phi2)))))) - ((0.5d0 - (0.5d0 * cos((2.0d0 * (0.5d0 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1))))))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * t_0) * t_0))), Math.sqrt(Math.abs(((0.5 + (0.5 * Math.cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * Math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (Math.cos(phi2) * Math.cos(phi1))))))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	return R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0))), math.sqrt(math.fabs(((0.5 + (0.5 * math.cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (math.cos(phi2) * math.cos(phi1))))))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(abs(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))) - Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))) * Float64(cos(phi2) * cos(phi1)))))))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	tmp = R * (2.0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(abs(((0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1))))))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Abs[N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 62.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\left|\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\right|}}}\right) \]
4. Add Preprocessing

Alternative 29: 62.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := \mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)\\ t_4 := t\_2 + \cos \phi_1 \cdot {\sin t\_0}^{2}\\ \mathbf{if}\;t\_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1 \leq 0.012:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* 0.5 (- lambda1 lambda2)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_3
         (fma
          (cos phi2)
          (* (cos phi1) (- 0.5 (* 0.5 (cos (* 2.0 t_0)))))
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))))
        (t_4 (+ t_2 (* (cos phi1) (pow (sin t_0) 2.0)))))
   (if (<= (+ t_2 (* (* (* (cos phi1) (cos phi2)) t_1) t_1)) 0.012)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 * (lambda1 - lambda2);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_3 = fma(cos(phi2), (cos(phi1) * (0.5 - (0.5 * cos((2.0 * t_0))))), (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))));
	double t_4 = t_2 + (cos(phi1) * pow(sin(t_0), 2.0));
	double tmp;
	if ((t_2 + (((cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.012) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 * Float64(lambda1 - lambda2))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_3 = fma(cos(phi2), Float64(cos(phi1) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_0))))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))))
	t_4 = Float64(t_2 + Float64(cos(phi1) * (sin(t_0) ^ 2.0)))
	tmp = 0.0
	if (Float64(t_2 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.012)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$2 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 0.012], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 0.012

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 52.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 52.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 50.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 50.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 0.012 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 57.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 57.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 30: 62.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\\ t_3 := t\_1 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;t\_1 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0 \leq 0.012:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_2
         (fma
          (fma -0.5 (cos (- lambda2 lambda1)) 0.5)
          (* (cos phi2) (cos phi1))
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5)))))))
        (t_3
         (+ t_1 (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))))
   (if (<= (+ t_1 (* (* (* (cos phi1) (cos phi2)) t_0) t_0)) 0.012)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (* (* (atan2 (sqrt t_2) (sqrt (- 1.0 t_2))) 2.0) R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_2 = fma(fma(-0.5, cos((lambda2 - lambda1)), 0.5), (cos(phi2) * cos(phi1)), (0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))))));
	double t_3 = t_1 + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double tmp;
	if ((t_1 + (((cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 0.012) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = (atan2(sqrt(t_2), sqrt((1.0 - t_2))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_2 = fma(fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5), Float64(cos(phi2) * cos(phi1)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5))))))
	t_3 = Float64(t_1 + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	tmp = 0.0
	if (Float64(t_1 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 0.012)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(Float64(atan(sqrt(t_2), sqrt(Float64(1.0 - t_2))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 0.012], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 0.012

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 52.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 52.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 50.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 50.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 0.012 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 57.4%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}} \cdot 2\right) \cdot R} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 31: 62.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_2 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\\ t_3 := t\_1 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;t\_1 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0 \leq 0.012:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_2
         (fma
          (cos phi1)
          (* (cos phi2) (fma -0.5 (cos (- lambda2 lambda1)) 0.5))
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5)))))))
        (t_3
         (+ t_1 (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))))
   (if (<= (+ t_1 (* (* (* (cos phi1) (cos phi2)) t_0) t_0)) 0.012)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_2 = fma(cos(phi1), (cos(phi2) * fma(-0.5, cos((lambda2 - lambda1)), 0.5)), (0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))))));
	double t_3 = t_1 + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double tmp;
	if ((t_1 + (((cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 0.012) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_2 = fma(cos(phi1), Float64(cos(phi2) * fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5))))))
	t_3 = Float64(t_1 + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	tmp = 0.0
	if (Float64(t_1 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 0.012)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 0.012], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 0.012

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 52.8

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 52.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 50.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 50.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 0.012 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 + \left(\mathsf{neg}\left(\phi_2\right)\right)}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} + \frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right) + \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\color{blue}{\sin \left(\frac{\phi_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\phi_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{\color{blue}{-2}}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \color{blue}{\cos \left(\frac{\phi_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{2}\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\phi_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 58.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_1 \cdot 0.5\right), \cos \left(\frac{\phi_2}{-2}\right), \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\frac{\phi_2}{-2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 57.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 32: 61.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (/
           (*
            (+ (cos (- phi2 phi1)) (cos (+ phi2 phi1)))
            (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
           2.0))))
   (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos((phi2 - phi1)) + cos((phi2 + phi1))) * (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2))))))) / 2.0);
	return R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos((phi2 - phi1)) + cos((phi2 + phi1))) * (0.5d0 - (0.5d0 * cos((2.0d0 * (0.5d0 * (lambda1 - lambda2))))))) / 2.0d0)
    code = r * (2.0d0 * atan2(sqrt(t_0), sqrt((1.0d0 - t_0))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((Math.cos((phi2 - phi1)) + Math.cos((phi2 + phi1))) * (0.5 - (0.5 * Math.cos((2.0 * (0.5 * (lambda1 - lambda2))))))) / 2.0);
	return R * (2.0 * Math.atan2(Math.sqrt(t_0), Math.sqrt((1.0 - t_0))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((math.cos((phi2 - phi1)) + math.cos((phi2 + phi1))) * (0.5 - (0.5 * math.cos((2.0 * (0.5 * (lambda1 - lambda2))))))) / 2.0)
	return R * (2.0 * math.atan2(math.sqrt(t_0), math.sqrt((1.0 - t_0))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(Float64(phi2 - phi1)) + cos(Float64(phi2 + phi1))) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))) / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos((phi2 - phi1)) + cos((phi2 + phi1))) * (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2))))))) / 2.0);
	tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. associate-*l* N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lift-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lift-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. cos-mult N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. associate-*l/ N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 60.2%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. associate-*l* N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

   5. lift-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

   6. lift-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

   7. cos-mult N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

   8. associate-*l/ N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

   9. lower-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

5. Applied rewrites 60.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}\right)}}\right) \]
6. Add Preprocessing

Alternative 33: 60.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_1 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi2)
          (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
          (pow (sin (* -0.5 phi2)) 2.0)))
        (t_1 (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))))
   (if (<= phi2 -1.6e-18)
     t_1
     (if (<= phi2 7.5e-20)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
            (pow (sin (* 0.5 phi1)) 2.0)))
          (sqrt
           (-
            1.0
            (fma
             (fma -0.5 (cos (- lambda2 lambda1)) 0.5)
             (cos phi1)
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))))))
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi2), (0.5 - (0.5 * cos((lambda1 - lambda2)))), pow(sin((-0.5 * phi2)), 2.0));
	double t_1 = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	double tmp;
	if (phi2 <= -1.6e-18) {
		tmp = t_1;
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((0.5 * phi1)), 2.0))), sqrt((1.0 - fma(fma(-0.5, cos((lambda2 - lambda1)), 0.5), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_1 = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
	tmp = 0.0
	if (phi2 <= -1.6e-18)
		tmp = t_1;
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(1.0 - fma(fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.6e-18], t$95$1, If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.6e-18 or 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

   5. Applied rewrites 60.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      3. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      9. lower-*.f64 44.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   8. Applied rewrites 44.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 44.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 44.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -1.6e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 34: 59.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\phi_2 \leq -1.6 \cdot 10^{-18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 7.5 \cdot 10^{-20}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (fma (cos (- lambda2 lambda1)) -0.5 0.5)
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_1
         (fma
          (cos phi2)
          (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
          (pow (sin (* -0.5 phi2)) 2.0)))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= phi2 -1.6e-18)
     t_2
     (if (<= phi2 7.5e-20)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi1), fma(cos((lambda2 - lambda1)), -0.5, 0.5), pow(sin((0.5 * phi1)), 2.0));
	double t_1 = fma(cos(phi2), (0.5 - (0.5 * cos((lambda1 - lambda2)))), pow(sin((-0.5 * phi2)), 2.0));
	double t_2 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double tmp;
	if (phi2 <= -1.6e-18) {
		tmp = t_2;
	} else if (phi2 <= 7.5e-20) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi1), fma(cos(Float64(lambda2 - lambda1)), -0.5, 0.5), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_1 = fma(cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	tmp = 0.0
	if (phi2 <= -1.6e-18)
		tmp = t_2;
	elseif (phi2 <= 7.5e-20)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.6e-18], t$95$2, If[LessEqual[phi2, 7.5e-20], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.6e-18 or 7.49999999999999981e-20 < phi2

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

   5. Applied rewrites 60.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      3. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      9. lower-*.f64 44.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   8. Applied rewrites 44.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. lower-*.f64 44.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 44.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -1.6e-18 < phi2 < 7.49999999999999981e-20

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. sqr-sin-a-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \color{blue}{\frac{1}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{-1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      11. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{-2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      12. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \frac{1}{-2} + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \color{blue}{\frac{1}{-2}}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 43.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       5. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       8. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       9. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \color{blue}{\frac{1}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{-1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       11. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{-2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       12. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \frac{1}{-2} + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       13. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \color{blue}{\frac{1}{-2}}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 43.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 35: 59.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (*
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
           (* (cos phi2) (cos phi1))))))
   (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1)));
	return R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + ((0.5d0 - (0.5d0 * cos((2.0d0 * (0.5d0 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1)))
    code = r * (2.0d0 * atan2(sqrt(t_0), sqrt((1.0d0 - t_0))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + ((0.5 - (0.5 * Math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (Math.cos(phi2) * Math.cos(phi1)));
	return R * (2.0 * Math.atan2(Math.sqrt(t_0), Math.sqrt((1.0 - t_0))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + ((0.5 - (0.5 * math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (math.cos(phi2) * math.cos(phi1)))
	return R * (2.0 * math.atan2(math.sqrt(t_0), math.sqrt((1.0 - t_0))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))) * Float64(cos(phi2) * cos(phi1))))
	return Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (sin(((phi1 - phi2) / 2.0)) ^ 2.0) + ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1)));
	tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. associate-*l* N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 59.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. associate-*l* N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

   4. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right)}}\right) \]

5. Applied rewrites 59.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}\right)}}\right) \]
6. Add Preprocessing

Alternative 36: 49.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \mathsf{fma}\left(\mathsf{fma}\left(-0.5, t\_0, 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_4 := \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(t\_0, -0.5, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;t\_2 \leq 0.15:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1
         (fma
          (fma -0.5 t_0 0.5)
          (cos phi1)
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* 0.5 (* 2.0 (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))))))
        (t_4 (fma (cos phi1) (fma t_0 -0.5 0.5) (pow (sin (* 0.5 phi1)) 2.0))))
   (if (<= t_2 -2e-60)
     (* (* (atan2 (sqrt t_1) (sqrt (- 1.0 t_1))) 2.0) R)
     (if (<= t_2 0.15)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = fma(fma(-0.5, t_0, 0.5), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (0.5 * (2.0 * (0.5 - (0.5 * cos((lambda1 - lambda2))))));
	double t_4 = fma(cos(phi1), fma(t_0, -0.5, 0.5), pow(sin((0.5 * phi1)), 2.0));
	double tmp;
	if (t_2 <= -2e-60) {
		tmp = (atan2(sqrt(t_1), sqrt((1.0 - t_1))) * 2.0) * R;
	} else if (t_2 <= 0.15) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = fma(fma(-0.5, t_0, 0.5), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(0.5 * Float64(2.0 * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))))))
	t_4 = fma(cos(phi1), fma(t_0, -0.5, 0.5), (sin(Float64(0.5 * phi1)) ^ 2.0))
	tmp = 0.0
	if (t_2 <= -2e-60)
		tmp = Float64(Float64(atan(sqrt(t_1), sqrt(Float64(1.0 - t_1))) * 2.0) * R);
	elseif (t_2 <= 0.15)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(-0.5 * t$95$0 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(0.5 * N[(2.0 * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[phi1], $MachinePrecision] * N[(t$95$0 * -0.5 + 0.5), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-60], N[(N[(N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[t$95$2, 0.15], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -1.9999999999999999e-60

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 42.6%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}} \cdot 2\right) \cdot R} \]

   if -1.9999999999999999e-60 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 0.149999999999999994

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

   5. Applied rewrites 60.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}\right)}}\right) \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \color{blue}{\left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      7. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      10. lower--.f64 50.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   8. Applied rewrites 50.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \color{blue}{\left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       6. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)}}\right) \]

       9. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       10. lower--.f64 48.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

   11. Applied rewrites 48.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

   12. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(2 \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 39.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(\color{blue}{0.5} - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

   15. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(2 \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]
   16. Step-by-step derivation

   17. Applied rewrites 35.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(\color{blue}{0.5} - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

   if 0.149999999999999994 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. sqr-sin-a-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \color{blue}{\frac{1}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{-1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      11. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{-2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      12. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \frac{1}{-2} + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \color{blue}{\frac{1}{-2}}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 43.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       5. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       8. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       9. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \color{blue}{\frac{1}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{-1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       11. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{-2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       12. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \frac{1}{-2} + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       13. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \color{blue}{\frac{1}{-2}}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 43.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 37: 49.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\\ t_3 := \mathsf{fma}\left(\mathsf{fma}\left(-0.5, t\_0, 0.5\right), \cos \phi_1, t\_2\right)\\ t_4 := t\_2 - \mathsf{fma}\left(t\_0, 0.5, -0.5\right) \cdot \cos \phi_1\\ t_5 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-60}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}} \cdot 2\right) \cdot R\\ \mathbf{elif}\;t\_1 \leq 0.15:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1))))))
        (t_3 (fma (fma -0.5 t_0 0.5) (cos phi1) t_2))
        (t_4 (- t_2 (* (fma t_0 0.5 -0.5) (cos phi1))))
        (t_5
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* 0.5 (* 2.0 (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))))))
   (if (<= t_1 -2e-60)
     (* (* (atan2 (sqrt t_3) (sqrt (- 1.0 t_3))) 2.0) R)
     (if (<= t_1 0.15)
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = 0.5 - (0.5 * cos((2.0 * (0.5 * phi1))));
	double t_3 = fma(fma(-0.5, t_0, 0.5), cos(phi1), t_2);
	double t_4 = t_2 - (fma(t_0, 0.5, -0.5) * cos(phi1));
	double t_5 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (0.5 * (2.0 * (0.5 - (0.5 * cos((lambda1 - lambda2))))));
	double tmp;
	if (t_1 <= -2e-60) {
		tmp = (atan2(sqrt(t_3), sqrt((1.0 - t_3))) * 2.0) * R;
	} else if (t_1 <= 0.15) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))
	t_3 = fma(fma(-0.5, t_0, 0.5), cos(phi1), t_2)
	t_4 = Float64(t_2 - Float64(fma(t_0, 0.5, -0.5) * cos(phi1)))
	t_5 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(0.5 * Float64(2.0 * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))))))
	tmp = 0.0
	if (t_1 <= -2e-60)
		tmp = Float64(Float64(atan(sqrt(t_3), sqrt(Float64(1.0 - t_3))) * 2.0) * R);
	elseif (t_1 <= 0.15)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-0.5 * t$95$0 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 - N[(N[(t$95$0 * 0.5 + -0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(0.5 * N[(2.0 * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-60], N[(N[(N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[t$95$1, 0.15], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -1.9999999999999999e-60

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 42.6%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}} \cdot 2\right) \cdot R} \]

   if -1.9999999999999999e-60 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 0.149999999999999994

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

   5. Applied rewrites 60.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}\right)}}\right) \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \color{blue}{\left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      7. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      10. lower--.f64 50.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   8. Applied rewrites 50.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \color{blue}{\left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       6. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)}}\right) \]

       9. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       10. lower--.f64 48.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

   11. Applied rewrites 48.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

   12. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(2 \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 39.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(\color{blue}{0.5} - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

   15. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(2 \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]
   16. Step-by-step derivation

   17. Applied rewrites 35.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(\color{blue}{0.5} - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

   if 0.149999999999999994 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 42.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right) - \color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right) \cdot \cos \phi_1}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 42.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right) - \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right) \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right) - \color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), 0.5, -0.5\right) \cdot \cos \phi_1}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 38: 49.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_3 := \left(\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \cdot 2\right) \cdot R\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-60}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0.15:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (fma -0.5 (cos (- lambda2 lambda1)) 0.5)
          (cos phi1)
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* 0.5 (* 2.0 (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))))))
        (t_3 (* (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))) 2.0) R)))
   (if (<= t_1 -2e-60)
     t_3
     (if (<= t_1 0.15)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(fma(-0.5, cos((lambda2 - lambda1)), 0.5), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (0.5 * (2.0 * (0.5 - (0.5 * cos((lambda1 - lambda2))))));
	double t_3 = (atan2(sqrt(t_0), sqrt((1.0 - t_0))) * 2.0) * R;
	double tmp;
	if (t_1 <= -2e-60) {
		tmp = t_3;
	} else if (t_1 <= 0.15) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(0.5 * Float64(2.0 * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))))))
	t_3 = Float64(Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))) * 2.0) * R)
	tmp = 0.0
	if (t_1 <= -2e-60)
		tmp = t_3;
	elseif (t_1 <= 0.15)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(0.5 * N[(2.0 * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-60], t$95$3, If[LessEqual[t$95$1, 0.15], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -1.9999999999999999e-60 or 0.149999999999999994 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 42.6%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}} \cdot 2\right) \cdot R} \]

   if -1.9999999999999999e-60 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 0.149999999999999994

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

   5. Applied rewrites 60.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}\right)}}\right) \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \color{blue}{\left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      7. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      10. lower--.f64 50.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   8. Applied rewrites 50.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \color{blue}{\left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       6. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)}}\right) \]

       9. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       10. lower--.f64 48.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

   11. Applied rewrites 48.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

   12. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(2 \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 39.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(\color{blue}{0.5} - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

   15. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(2 \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]
   16. Step-by-step derivation

   17. Applied rewrites 35.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(\color{blue}{0.5} - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 39: 40.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R\\ \mathbf{if}\;t\_1 \leq -0.39:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.215:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* 0.5 (* 2.0 (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (*
          (*
           (atan2
            (sqrt
             (fma
              (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
              (cos phi1)
              (* (* phi1 phi1) 0.25)))
            (sqrt
             (-
              1.0
              (fma
               (cos phi2)
               (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
               (pow (sin (* -0.5 phi2)) 2.0)))))
           2.0)
          R)))
   (if (<= t_1 -0.39)
     t_2
     (if (<= t_1 0.215)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (0.5 * (2.0 * (0.5 - (0.5 * cos((lambda1 - lambda2))))));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = (atan2(sqrt(fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), ((phi1 * phi1) * 0.25))), sqrt((1.0 - fma(cos(phi2), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((-0.5 * phi2)), 2.0))))) * 2.0) * R;
	double tmp;
	if (t_1 <= -0.39) {
		tmp = t_2;
	} else if (t_1 <= 0.215) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(0.5 * Float64(2.0 * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))))))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), Float64(Float64(phi1 * phi1) * 0.25))), sqrt(Float64(1.0 - fma(cos(phi2), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(-0.5 * phi2)) ^ 2.0))))) * 2.0) * R)
	tmp = 0.0
	if (t_1 <= -0.39)
		tmp = t_2;
	elseif (t_1 <= 0.215)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(0.5 * N[(2.0 * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[t$95$1, -0.39], t$95$2, If[LessEqual[t$95$1, 0.215], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.39000000000000001 or 0.214999999999999997 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.3

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 21.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 21.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.2%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]
   17. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       3. lower-cos.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       4. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       5. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       6. lower-*.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       7. lower--.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       8. lower-pow.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       9. lower-sin.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

       10. lower-*.f64 29.0

           \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}} \cdot 2\right) \cdot R \]

   18. Applied rewrites 29.0%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}} \cdot 2\right) \cdot R \]

   if -0.39000000000000001 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 0.214999999999999997

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

   5. Applied rewrites 60.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}\right)}}\right) \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \color{blue}{\left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      7. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      10. lower--.f64 50.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   8. Applied rewrites 50.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \color{blue}{\left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       6. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)}}\right) \]

       9. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       10. lower--.f64 48.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

   11. Applied rewrites 48.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

   12. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(2 \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 39.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(\color{blue}{0.5} - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

   15. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(2 \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]
   16. Step-by-step derivation

   17. Applied rewrites 35.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(\color{blue}{0.5} - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 40: 37.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_1 := \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\\ \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq 0.215:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, t\_1\right)}}{e^{\log \left({\left(1 - \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), \cos \phi_1, t\_1\right)\right)}^{2}\right) \cdot 0.25}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* 0.5 (* 2.0 (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))))))
        (t_1 (* (* phi1 phi1) 0.25)))
   (if (<= (sin (/ (- lambda1 lambda2) 2.0)) 0.215)
     (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
     (*
      (*
       (atan2
        (sqrt
         (fma
          (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
          (cos phi1)
          t_1))
        (exp
         (*
          (log
           (pow
            (-
             1.0
             (fma (fma (cos (- lambda2 lambda1)) -0.5 0.5) (cos phi1) t_1))
            2.0))
          0.25)))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (0.5 * (2.0 * (0.5 - (0.5 * cos((lambda1 - lambda2))))));
	double t_1 = (phi1 * phi1) * 0.25;
	double tmp;
	if (sin(((lambda1 - lambda2) / 2.0)) <= 0.215) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = (atan2(sqrt(fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), t_1)), exp((log(pow((1.0 - fma(fma(cos((lambda2 - lambda1)), -0.5, 0.5), cos(phi1), t_1)), 2.0)) * 0.25))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(0.5 * Float64(2.0 * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))))))
	t_1 = Float64(Float64(phi1 * phi1) * 0.25)
	tmp = 0.0
	if (sin(Float64(Float64(lambda1 - lambda2) / 2.0)) <= 0.215)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), t_1)), exp(Float64(log((Float64(1.0 - fma(fma(cos(Float64(lambda2 - lambda1)), -0.5, 0.5), cos(phi1), t_1)) ^ 2.0)) * 0.25))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(0.5 * N[(2.0 * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]}, If[LessEqual[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 0.215], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[Log[N[Power[N[(1.0 - N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 0.214999999999999997

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      7. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2}} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      8. associate-*l/ N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

      9. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}{2}}\right)}}\right) \]

   5. Applied rewrites 60.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}}\right)}}\right) \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \color{blue}{\left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      6. lower-neg.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      7. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

      10. lower--.f64 50.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   8. Applied rewrites 50.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{\left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}{2}\right)}}\right) \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \color{blue}{\left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)}}\right) \]

       3. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(\mathsf{neg}\left(\phi_1\right)\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       6. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)}}\right) \]

       9. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

       10. lower--.f64 48.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

   11. Applied rewrites 48.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}\right)}}\right) \]

   12. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(2 \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 39.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(\color{blue}{0.5} - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(\left(\cos \phi_1 + \cos \left(-\phi_1\right)\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

   15. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \frac{1}{2} \cdot \left(2 \cdot \left(\color{blue}{\frac{1}{2}} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]
   16. Step-by-step derivation

   17. Applied rewrites 35.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + 0.5 \cdot \left(2 \cdot \left(\color{blue}{0.5} - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)}}\right) \]

   if 0.214999999999999997 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-pow.f64 31.3

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

       2. lower-pow.f64 21.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   13. Applied rewrites 21.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

   15. Applied rewrites 19.2%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]
   16. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}} \cdot 2\right) \cdot R \]

       2. pow1/2 N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\frac{1}{2}}}} \cdot 2\right) \cdot R \]

       3. metadata-eval N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{{\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\color{blue}{\left(\frac{1}{4} + \frac{1}{4}\right)}}} \cdot 2\right) \cdot R \]

       4. pow-prod-up N/A

          \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\frac{1}{4}} \cdot {\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\frac{1}{4}}}} \cdot 2\right) \cdot R \]

   17. Applied rewrites 27.5%

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\color{blue}{e^{\log \left({\left(1 - \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)\right)}^{2}\right) \cdot 0.25}}} \cdot 2\right) \cdot R \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 41: 27.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\\ \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, t\_0\right)}}{e^{\log \left({\left(1 - \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), \cos \phi_1, t\_0\right)\right)}^{2}\right) \cdot 0.25}} \cdot 2\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (* phi1 phi1) 0.25)))
   (*
    (*
     (atan2
      (sqrt
       (fma (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5)) (cos phi1) t_0))
      (exp
       (*
        (log
         (pow
          (- 1.0 (fma (fma (cos (- lambda2 lambda1)) -0.5 0.5) (cos phi1) t_0))
          2.0))
        0.25)))
     2.0)
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (phi1 * phi1) * 0.25;
	return (atan2(sqrt(fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), t_0)), exp((log(pow((1.0 - fma(fma(cos((lambda2 - lambda1)), -0.5, 0.5), cos(phi1), t_0)), 2.0)) * 0.25))) * 2.0) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(phi1 * phi1) * 0.25)
	return Float64(Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), t_0)), exp(Float64(log((Float64(1.0 - fma(fma(cos(Float64(lambda2 - lambda1)), -0.5, 0.5), cos(phi1), t_0)) ^ 2.0)) * 0.25))) * 2.0) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]}, N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[Log[N[Power[N[(1.0 - N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.3

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.3%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. lower-*.f64 46.4

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.4%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-pow.f64 31.3

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

    2. lower-pow.f64 21.8

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

13. Applied rewrites 21.8%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

15. Applied rewrites 19.2%

    \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]
16. Step-by-step derivation

    1. lift-sqrt.f64 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}} \cdot 2\right) \cdot R \]

    2. pow1/2 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\frac{1}{2}}}} \cdot 2\right) \cdot R \]

    3. metadata-eval N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{{\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\color{blue}{\left(\frac{1}{4} + \frac{1}{4}\right)}}} \cdot 2\right) \cdot R \]

    4. pow-prod-up N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\frac{1}{4}} \cdot {\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\frac{1}{4}}}} \cdot 2\right) \cdot R \]

17. Applied rewrites 27.5%

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\color{blue}{e^{\log \left({\left(1 - \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)\right)}^{2}\right) \cdot 0.25}}} \cdot 2\right) \cdot R \]
18. Add Preprocessing

Alternative 42: 27.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\\ \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, t\_0\right)}}{{\left({\left(1 - \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), \cos \phi_1, t\_0\right)\right)}^{2}\right)}^{0.25}} \cdot 2\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (* phi1 phi1) 0.25)))
   (*
    (*
     (atan2
      (sqrt
       (fma (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5)) (cos phi1) t_0))
      (pow
       (pow
        (- 1.0 (fma (fma (cos (- lambda2 lambda1)) -0.5 0.5) (cos phi1) t_0))
        2.0)
       0.25))
     2.0)
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (phi1 * phi1) * 0.25;
	return (atan2(sqrt(fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), t_0)), pow(pow((1.0 - fma(fma(cos((lambda2 - lambda1)), -0.5, 0.5), cos(phi1), t_0)), 2.0), 0.25)) * 2.0) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(phi1 * phi1) * 0.25)
	return Float64(Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), t_0)), ((Float64(1.0 - fma(fma(cos(Float64(lambda2 - lambda1)), -0.5, 0.5), cos(phi1), t_0)) ^ 2.0) ^ 0.25)) * 2.0) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]}, N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Power[N[Power[N[(1.0 - N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 0.25], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.3

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.3%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. lower-*.f64 46.4

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.4%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-pow.f64 31.3

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

    2. lower-pow.f64 21.8

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

13. Applied rewrites 21.8%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

15. Applied rewrites 19.2%

    \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]
16. Step-by-step derivation

    1. lift-sqrt.f64 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}} \cdot 2\right) \cdot R \]

    2. pow1/2 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\frac{1}{2}}}} \cdot 2\right) \cdot R \]

    3. metadata-eval N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{{\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\color{blue}{\left(\frac{1}{4} + \frac{1}{4}\right)}}} \cdot 2\right) \cdot R \]

    4. pow-prod-up N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\frac{1}{4}} \cdot {\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\frac{1}{4}}}} \cdot 2\right) \cdot R \]

17. Applied rewrites 27.5%

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\color{blue}{{\left({\left(1 - \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)\right)}^{2}\right)}^{0.25}}} \cdot 2\right) \cdot R \]
18. Add Preprocessing

Alternative 43: 24.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\\ \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, t\_0\right)}}{\frac{1}{{\left(1 - \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), \cos \phi_1, t\_0\right)\right)}^{-0.5}}} \cdot 2\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (* phi1 phi1) 0.25)))
   (*
    (*
     (atan2
      (sqrt
       (fma (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5)) (cos phi1) t_0))
      (/
       1.0
       (pow
        (- 1.0 (fma (fma (cos (- lambda2 lambda1)) -0.5 0.5) (cos phi1) t_0))
        -0.5)))
     2.0)
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (phi1 * phi1) * 0.25;
	return (atan2(sqrt(fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), t_0)), (1.0 / pow((1.0 - fma(fma(cos((lambda2 - lambda1)), -0.5, 0.5), cos(phi1), t_0)), -0.5))) * 2.0) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(phi1 * phi1) * 0.25)
	return Float64(Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), t_0)), Float64(1.0 / (Float64(1.0 - fma(fma(cos(Float64(lambda2 - lambda1)), -0.5, 0.5), cos(phi1), t_0)) ^ -0.5))) * 2.0) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]}, N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / N[(1.0 / N[Power[N[(1.0 - N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.3

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.3%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. lower-*.f64 46.4

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.4%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-pow.f64 31.3

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

    2. lower-pow.f64 21.8

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

13. Applied rewrites 21.8%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

15. Applied rewrites 19.2%

    \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]
16. Step-by-step derivation

    1. lift-sqrt.f64 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}} \cdot 2\right) \cdot R \]

    2. pow1/2 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\frac{1}{2}}}} \cdot 2\right) \cdot R \]

    3. metadata-eval N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{{\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{2}\right)\right)}}} \cdot 2\right) \cdot R \]

    4. pow-neg N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\color{blue}{\frac{1}{{\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\frac{-1}{2}}}}} \cdot 2\right) \cdot R \]

17. Applied rewrites 24.3%

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\color{blue}{\frac{1}{{\left(1 - \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)\right)}^{-0.5}}}} \cdot 2\right) \cdot R \]
18. Add Preprocessing

Alternative 44: 24.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\\ \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, t\_0\right)}}{{\left(1 - \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), \cos \phi_1, t\_0\right)\right)}^{0.5}} \cdot 2\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (* phi1 phi1) 0.25)))
   (*
    (*
     (atan2
      (sqrt
       (fma (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5)) (cos phi1) t_0))
      (pow
       (- 1.0 (fma (fma (cos (- lambda2 lambda1)) -0.5 0.5) (cos phi1) t_0))
       0.5))
     2.0)
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (phi1 * phi1) * 0.25;
	return (atan2(sqrt(fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), cos(phi1), t_0)), pow((1.0 - fma(fma(cos((lambda2 - lambda1)), -0.5, 0.5), cos(phi1), t_0)), 0.5)) * 2.0) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(phi1 * phi1) * 0.25)
	return Float64(Float64(atan(sqrt(fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), cos(phi1), t_0)), (Float64(1.0 - fma(fma(cos(Float64(lambda2 - lambda1)), -0.5, 0.5), cos(phi1), t_0)) ^ 0.5)) * 2.0) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]}, N[(N[(N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Power[N[(1.0 - N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.3

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.3%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. lower-*.f64 46.4

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.4%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-pow.f64 31.3

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

    2. lower-pow.f64 21.8

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

13. Applied rewrites 21.8%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

15. Applied rewrites 19.2%

    \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]
16. Step-by-step derivation

    1. lift-sqrt.f64 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}} \cdot 2\right) \cdot R \]

    2. pow1/2 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)\right)}^{\frac{1}{2}}}} \cdot 2\right) \cdot R \]

    3. lower-pow.f64 24.3

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)\right)}^{0.5}}} \cdot 2\right) \cdot R \]

17. Applied rewrites 24.3%

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)\right)}^{0.5}}} \cdot 2\right) \cdot R \]
18. Add Preprocessing

Alternative 45: 19.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)\\ \left(\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \cdot 2\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
          (+ 1.0 (* -0.5 (pow phi1 2.0)))
          (* (* phi1 phi1) 0.25))))
   (* (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))) 2.0) R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), (1.0 + (-0.5 * pow(phi1, 2.0))), ((phi1 * phi1) * 0.25));
	return (atan2(sqrt(t_0), sqrt((1.0 - t_0))) * 2.0) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))), Float64(Float64(phi1 * phi1) * 0.25))
	return Float64(Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))) * 2.0) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 * phi1), $MachinePrecision] * 0.25), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Initial program 62.3%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.3

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.3%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. lower-*.f64 46.4

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.4%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   2. lower-pow.f64 31.3

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]
12. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{4} \cdot {\phi_1}^{2}\right)}}\right) \]

    2. lower-pow.f64 21.8

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

13. Applied rewrites 21.8%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.25 \cdot {\phi_1}^{2}\right)}}\right) \]

15. Applied rewrites 19.2%

    \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R} \]

16. Taylor expanded in phi1 around 0

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \color{blue}{\frac{-1}{2} \cdot {\phi_1}^{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]
17. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot \color{blue}{{\phi_1}^{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{\color{blue}{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

    3. lower-pow.f64 19.2

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R \]

18. Applied rewrites 19.2%

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + \color{blue}{-0.5 \cdot {\phi_1}^{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_1, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R \]

19. Taylor expanded in phi1 around 0

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \color{blue}{\frac{-1}{2} \cdot {\phi_1}^{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]
20. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot \color{blue}{{\phi_1}^{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \frac{1}{2}, 1 + \frac{-1}{2} \cdot {\phi_1}^{\color{blue}{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot \frac{1}{4}\right)}} \cdot 2\right) \cdot R \]

    3. lower-pow.f64 19.2

       \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R \]

21. Applied rewrites 19.2%

    \[\leadsto \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + -0.5 \cdot {\phi_1}^{2}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, 1 + \color{blue}{-0.5 \cdot {\phi_1}^{2}}, \left(\phi_1 \cdot \phi_1\right) \cdot 0.25\right)}} \cdot 2\right) \cdot R \]
22. Add Preprocessing

Reproduce

herbie shell --seed 2025181 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* R (* 2.0 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))) (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))))))))